MONODROMY AND HE´ NON MAPPINGS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Christopher Lipa August 2009 c 2009 Christopher Lipa ALL RIGHTS RESERVED MONODROMY AND HE´ NON MAPPINGS Christopher Lipa, Ph.D. Cornell University 2009 We discuss the monodromy action of loops in the horseshoe locus of the He´non map on its Julia set. We will show that for a particular class of loops there is a certain combinatorially-defined subset of the He´non Julia set which must remain invariant under the monodromy action of loops in certain regions. We will then describe a conjecture for what the monodromy actions of these loops are as well as a possible connection between the algebraic structure of automorphisms of the full 2-shift and the existence of certain types of loops in the horseshoe locus. BIOGRAPHICAL SKETCH Christopher Lipa graduated from North Carolina State University in 2003 where he read mathematics and computer science. He attended graduate school at Cornell University, graduating in 2009 with a Ph. D. in mathematics. iii This thesis is dedicated to my parents. iv ACKNOWLEDGEMENTS This work depends on fundamental insights from Sarah Koch, John Milnor, Adrien Douady, and John Hubbard. The programs SaddleDrop and FractalAsm written by Karl Papadantonakis were essential to the discovery of the phenomenon that this thesis describes. I’d like to express gratitude for conversations with John Smillie, Dierk Schleicher, and Laurent Bartholdi. I also wish to thank Zin Arai, William Thurston, Eric Bedford, and Ralph Oberste-Vorth. This work also could not have been possible if not for the generous financial support of the Mathematics Department of Cornell University and the National Science Foundation’s VIGRE Grant. v TABLE OF CONTENTS Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii iv v vi viii 1 Introduction 1.1 He´non Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Monodromy Image Conjecture . . . . . . . . . . . . . . . . . . . . . . 1.3 Monodromy Action Conjectures . . . . . . . . . . . . . . . . . . . . . 1.4 Structurally Stable Set . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 3 5 2 Preliminaries 7 2.1 Standard Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Monodromy Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Monodromy in the HOV region 11 3.1 Inverse Limit Description . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Homotopy of HOV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Monodromy Action of γb . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Monodromy Action of γc . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Orbit Portraits and Puzzles 13 4.1 Formal Orbit Portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Actual Orbit Portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Puzzle Pieces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.4 Fattened Puzzle Pieces . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.5 Itineraries Relative to Orbit Portraits . . . . . . . . . . . . . . . . . . . 21 4.6 Kneading Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.6.1 Kneading Sequences of Quadratic Polynomials . . . . . . . . . 21 4.6.2 Kneading Sequences of Orbit Portraits . . . . . . . . . . . . . . 24 4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.7.1 The Airplane . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.7.2 BABB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5 XWc 36 5.1 Defining XWc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Multi-Itineraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5.3 Adaptation to Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.4 Relations Between Points and Itineraries . . . . . . . . . . . . . . . . . 42 5.5 Continuity of XWc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.6 W-itineraries of XWc . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 vi 6 XWb,c 52 6.1 Crossed Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 6.2 Horizontal Disk Contraction . . . . . . . . . . . . . . . . . . . . . . . 55 6.3 Perturbations of One-Dimensional Orbit Portraits . . . . . . . . . . . . 57 6.4 Continuity of XWb,c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.5 Coding XWb,c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.6 Relationships Between Points and Itineraries . . . . . . . . . . . . . . . 68 7 Monodromy Invariant 70 8 Monodromy Conjectures 72 8.1 Speculative Structure of He´non Parameter Space . . . . . . . . . . . . 72 8.2 Monodromy Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9 Examples 9.1 B BAA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 BB BAA and AB BAA . . . . . . . . . . . . . . . . . . . . . . . . 9.3 A BAA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 A BAA, A BABBA . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 A BAA, A BABBA, B BAA . . . . . . . . . . . . . . . . . . . . . 9.6 ABAAB BAA and BBAAB BAA . . . . . . . . . . . . . . . . . . . 76 77 86 87 87 91 96 A Monodromies of Inverse Limit Systems 100 A.1 Inverse Limit System Setup . . . . . . . . . . . . . . . . . . . . . . . . 100 A.2 Coding Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 A.3 Monodromy Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Bibliography 108 vii LIST OF FIGURES 4.1 M with 3/7 and 4/7 parameter rays. . . . . . . . . . . . . . . . . . . . 27 4.2 Airplane polynomial with actual orbit portrait Oair . . . . . . . . . . . 28 4.3 Oair and the Julia set of airplane polynomial . . . . . . . . . . . . . . . 29 4.4 Γair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.5 Mandelbrot set with 13/31 and 18/31 parameter rays . . . . . . . . . . 32 4.6 OBABB and the Julia set of fBABB . . . . . . . . . . . . . . . . . . . . . 33 4.7 Puzzle pieces for fBABB associated with PBABB . . . . . . . . . . . . . . 34 4.8 ΓBABB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9.1 Parameter slice with b = 0 . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Parameter slice with b = 0.005i . . . . . . . . . . . . . . . . . . . . . 9.3 Parameter slice with b = 0.01i . . . . . . . . . . . . . . . . . . . . . . 9.4 Parameter slice with b = 0.015i . . . . . . . . . . . . . . . . . . . . . 9.5 Parameter slice with b = 0.02i . . . . . . . . . . . . . . . . . . . . . . 9.6 Parameter slice with b = 0.03i . . . . . . . . . . . . . . . . . . . . . . 9.7 Parameter slice with b = 0.05i . . . . . . . . . . . . . . . . . . . . . . 9.8 Loop around B herd of Wair with b = 0.05i . . . . . . . . . . . . . . . 9.9 Loops around BB and AB herds of Wair with b = 0.2 + 0.3i . . . . . . 9.10 Loop around A herd of WBABB . . . . . . . . . . . . . . . . . . . . . 9.11 Parameter slice with b = −0.03 + 0.02i . . . . . . . . . . . . . . . . . 9.12 Parameter slice with b = −0.03 + 0.01i . . . . . . . . . . . . . . . . . 9.13 Parameter slice with b = −0.03 . . . . . . . . . . . . . . . . . . . . . 9.14 ABAAB and BBAAB herds of Wair with b = −0.1 + 0.9i . . . . . . . . 9.15 Other herds near the ABAAB and BBAAB herds of Wair with b = −0.1+ 0.9i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.16 Other herds obstructing loops around ABAAB and BBAAB herds of Wair with b = −0.1 + 0.9i . . . . . . . . . . . . . . . . . . . . . . . . 78 79 80 81 82 83 84 85 86 88 92 93 94 97 98 99 viii CHAPTER 1 INTRODUCTION 1.1 He´non Mappings In 1963, Lorentz [Lor63] introduced a three-dimensional differential equation which was an attempt at a simplified model of convection of air currents in the atmosphere. There is a particular Poincare´ first-return map that He´non [He´n76] noticed had an action that is qualitatively similar to, but not exactly equal to the two parameter polynomial diffeomorphism of the plane, now called the He´non map:    Hb,c :  x y  →  x2 + c − by x  Over the past four decades, the He´non map has arguably been the most-studied multidimensional dynamical system. This is in part due to the fact that the He´non mapping is a perturbation of the (mostly) well-understood one-dimensional logistic family and has a relatively simple formulation, yet the dynamics of the He´non map are fantastically complicated and the He´non map exhibits chaotic phenomena that do not appear in onedimensional maps. Subsequent to the success realized in understanding logistic maps by complexifying and bringing complex analytic techniques to bear, in the 1980s, Hubbard [Hub86] had the idea to complexify the He´non mapping and examine structures in complex dynamical and parameter space in order to try to glean insight into the real mappings. One result revealed through the work of Hubbard and Oberste-Vorth [HOV94a] is that there is a large region of parameter space, called the horseshoe locus, where the dynamics on the Julia set is hyperbolic and conjugate to Smale’s horseshoe map. 1 Arai [Ara08] has more recently exhibited loops in this horseshoe locus in addition to the “obvious” classes of loops exhibited in Hubbard and Oberste-Vorth’s work. If one has a loop in the horseshoe locus, one can continuously follow points of the Julia set around and back to some (possibly different) point in the Julia set at the basepoint. This induces an action on the Julia set at the basepoint of the loop, which is called the monodromy action associated with the loop. The monodromy action maps a loop to a continuous automorphism of the Julia set of the basepoint of the loop, and this automorphism must commute with the action of the He´non map. We call the image of the monodromy action the induced monodromy group, and (up to conjugacy) the monodromy group is independent of the basepoint of the loop in path-connected regions of the horseshoe locus. From the monodromy action of a loop in parameter space, one can deduce implications on what types of dynamics must occur as the loop is homotoped to a constant, and we hope that further results may use monodromy as one of many tools in developing a road map of He´non parameter space similar to Douady, Hubbard, Schleicher, and Milnor’s combinatorial description of quadratic polynomial parameter space. 1.2 Monodromy Image Conjecture In the complement of the Mandelbrot set, the Julia set is hyperbolic and isomorphic to the one-sided shift on sequences of two symbols. Aut Σ+2 , σ is generated by the automorphism that acts on sequences by exchanging A and B. The generator of the fundamental group of the complement of the Mandelbrot set induces this automorphism on the Julia set at any base point. In other words, the induced monodromy group of the shift locus of quadratic polynomials is Aut Σ+2 , σ . For degree d one-complex-dimensional polynomial maps, there is also a shift locus 2 L in parameter space, where the polynomial restricted to the Julia set is hyperbolic and conjugate to the one-sided shift on d symbols. Loops in L based at a specific basepoint also have a continuous monodromy action on Σ+d which commutes with the shift. There is again a natural monodromy action π1(L) → Aut(Σ+d , σ). The situation is here is more complicated, as whenever d > 2, then Aut(Σ+d , σ) is infinitely generated. However, as Blanchard, Devaney, and Keen [BDK91] show, the induced monodromy group of the shift locus is again Aut(Σ+d , σ), and moreover, Blanchard, Devaney, and Keen give an explicit method to realize the generators. Based on these facts, Hubbard conjectured that the pattern continued in the twodimensional case. Conjecture 1.1 (Hubbard). The induced monodromy group of the horseshoe locus together with the shift generate Aut(Σ2, σ). The one-sided shifts are relatively well-understood. By contrast, the group of continuous automorphisms of the two-sided shift on two symbols which commute with the shift is not [BLR88]. We know it contains as subgroups all finite groups as well as Z and the product of countably many Z’s. It also contains a subgroup isomorphic to the free group on infinitely many generators. It is unknown if the group is generated by involutions and the shift. No nontrivial generating set is known. Proving or refuting Conjecture 1.1 would be a significant advance for automata theory as well as dynamics. 1.3 Monodromy Action Conjectures The Mandelbrot set lives in a complex slice of He´non parameter space. Koch ([Koc05] and [Koc07]) experimentally found that components of the Mandelbrot set bifurcate in 3 He´non parameter space into many pieces, called herds. All of the interesting loops in the horseshoe locus that are presently known wrap around these herds. Extensive computer experimentation has led to Conjecture 8.3, which describes what the Monodromy action is for this class of loops which wrap around Koch’s herds. Conjecturally, non-hyperbolic components in parameter space can be labeled with a finite string on two symbols, by which herd they are in and can also be followed back to a region of the Mandelbrot set. Conjecture 8.3 states that the monodromy action of a loop around such a non-hyperbolic component in parameter space is described by a natural generalization of marker automorphisms, which we call compound marker automorphisms (defined in Chapter 8). We postulate that the compound marker automorphism describing the monodromy action for such a loop has two parts. Our conjecture is that the prefix of the marker string comes from the labeling of which herd the looped component is in and that the suffix of the marker string comes from the kneading sequences realized by the polynomials in the region of the Mandelbrot set where the looped component can be followed back to as the Jacobian moves to 0. An example illustrating Conjecture 8.3 is that experimentally we have found that the monodromy action of a loop around the B-herd of the region further out than the airplane polynomial in the natural ordering of the Mandelbrot set to be described by the marker automorphism B BAA. We conjecture that the B that comes before the corresponds to the fact the loop goes around the B herd and that the BAA coming after the corresponds to the fact that every polynomial that is further out than the airplane in the natural ordering on the Mandelbrot set has a kneading sequence with initial segment BAA. It is possible to construct marker strings which do not yield automorphims. Conjecture 8.4 states that when this occurs, there is some obstruction in parameter space to 4 loops going around the prescribed herds and only the prescribed herds. 1.4 Structurally Stable Set If Conjecture 8.3 is correct, then loops around non-hyperbolic components coming from any particular region of the Mandelbrot set must have a trivial action on all points of the Julia set that lack a particular coding that relates to the symbolic dynamics present in that region. Though we don’t prove Conjecture 8.3, we do prove this consequence of the conjecture in Theorem 7.5, which states that the monodromy action must be invariant on the points in the Julia set that lack a particular (finite) list of words in their symbolic coding. To this end, we use two powerful tools. The first is Milnor’s orbit portrait construction in one complex variable dynamics which is described in [Mil00] and was inspired by the works of Douady, Hubbard, and Schleicher ([DH82], [DH84], [DH85], [Sch94], [Sch00], and [Sch04]). Milnor’s construction gives a puzzle decomposition of onecomplex-dimensional dynamical space. With every orbit portrait, we get a Markov graph Γ that describes the allowable transitions between puzzle pieces. Most importantly, we get expansion on all of the puzzle pieces that don’t include the critical point. Hence, the set of points XWc (defined in Chapter 5 as the set of points which never visit this critical puzzle piece) is hyperbolic and stable under small perterbation. Also, we get a two-symbol coding of this set described by Corollary 5.22, which states that all possible two-symbol codings are realizable by points of XWc , with the exception that there is a finite list of finite strings which may only appear at the beginning of a coding. This finite list of strings depends only on the abstract orbit portrait from which the puzzle pieces were generated and is closely related to the kneading sequences of 5 one-dimensional polynomials that satisfy the orbit portrait. The second major tool we use is Hubbard and Oberste-Vorth’s crossed mappings, as described in [HOV94b]. Roughly speaking, a crossed mapping is a map from one bidisk over another with contraction in one direction and expansion in the other. Hubbard and Oberste-Vorth show that if we have a bi-infinite sequence of degree-one crossed maps, then there is precisely one point who visits each bi-disk in turn. In Chapter 6, we construct two-dimensional puzzle pieces which are extensions in the y-direction of the one-dimensional puzzle pieces from Chapter 4. Then the mapping of one puzzle piece over another in one-variable dynamics implies that there is a 1-crossed mapping of the corresponding two-dimensional puzzle pieces. Any bi-infinite path in Γ then yields a point of the He´non Julia set. This construction works in a neighborhood of any one-dimensional quadratic wake living inside He´non parameter space, so this subset of the Julia set forms a trivial fibre-bundle, and we see in Chapter 7 that any loop in this region must have a trivial monodromy action on this subset. This subset corresponds to precisely the points whose itinerary avoids the critical puzzle piece. The points which avoid the critical puzzle piece are those which do not have in their itineraries the initial kneading sequences of polynomials in that region of the Mandelbrot set. Theorem 7.5 expresses the fact that there must be a trivial monodromy action on this combinatorially-defined subset in a domain around the onedimensional wake. 6 CHAPTER 2 PRELIMINARIES 2.1 Standard Definitions For the relevant definitions, we follow [HOV94a], [BS06], and [Ara08]. For parameter values b, c ∈ C, we define the He´non map Hb,c : C2 → C2 by:    Hb,c :  x y  →  x2 + c − by x  When b = 0, the first coordinate reduces to a quadratic polynomial on C. When b 0, the map is diffeomorphism of C2. We define the following dynamically meaningful subsets of C2:    Kb±,c =  x y  lim n→∞ Ha◦,±cn  x y   ∞  as well as: Ub±,c = C2 \ Kb±,c Jb±,c = ∂Kb±,c = ∂Ub±,c Ub,c = Ub+,c ∩ Ub−,c Kb,c = Kb+,c ∩ Kb−,c Jb,c = Jb+,c ∩ Jb−,c KbR,c = Kb,c ∩ R2 JbR,c = Jb,c ∩ R2 7 We also define the Green’s function G+b,c : C2 → R by:      G+b,c  x y  = lim n→∞ 1 2n log pr1Hb◦,nc  x y  where pr1 is projection to the first co-ordinate. For any parameter value (b, c), there exists an Rb,c ∈ R+ so that dynamical space is partitioned into three regions: Vb0,c = (x, y) |x| ≤ Rb,c and |y| ≤ Rb,c Vb−,c = (x, y) |y| ≥ |x| and |y| ≥ Rb,c Vb+,c = (x, y) |x| ≥ |y| and |x| ≥ Rb,c with the dynamics on these sets such that Hb,c(Vb+,c) ⊂ Vb+,c ⊂ Ub+,c and Hb−,1c(Vb−,c) ⊂ Vb−,c ⊂ Ub−,c. We define the complex horseshoe locus as the following region in parameter space: HC = (b, c) ∈ C2 Hb,c|Kb,c is hyperbolic and conjugate to the horseshoe and we define the real horseshoe locus as the following region in parameter space: HR = (b, c) ∈ R2 H |b,c KbR,c is hyperbolic and conjugate to the horseshoe Here, the horseshoe refers to the space of bi-infinite sequences on two symbols under the action of the shift map (also referred to as the full 2-shift). Let HOV = (b, c) ∈ C2 b 0 and |c| > 2(1 + |b|)2 , the Hubbard-Oberste-Vorth region described in [OV87], which is a connected subset of HC. Let H0C be the connected component of HC which contains HOV. It is unknown if HC = H0C. 8 Also define: Conn = (b, c) ∈ C2 Jb,c is connected Let fc(z) = z2 + c. Let M denote the Mandelbrot set. Let A and B be hyperbolic components of M with the parameter rays with angles θA− and θA+ landing at the root point of A and the parameter rays with angles θB− and θB+ landing at the root point of B. We define a partial ordering ≺ on hyperbolic components as follows: A ≺ B if any only if (θB−, θB+) ⊂ (θA− , θA+ ). Also define the Green’s function G : C → R+0 associated with fc to be: G(z) = lim n→∞ 1 2n log+ fc◦n(z) Let Σ2 denote the two-sided shift on two symbols. Let Σ+2 denote the one-sided shift on two symbols, let σ denote the shift operator on each of these spaces, and let δ denote the automorphism of each of these spaces that acts on sequences by exchanging the two symbols. For any dynamical system f : M → M on a topological space, let Aut(M, f ) denote the continuous automorphisms of M that commute with f . Also let l←i−m−(M, f ) denote the inverse limit system: l←i−m−(M, f ) = (. . . , x−1, x0, x1, . . .) xi ∈ M and f (xi) = xi+1 for all i ∈ Z Define b0 = 1 100 , c = −3, and p0 = (b0, c0). Then p0 ∈ H OV ∩ HR. Also there is a canonical homeomorphism from Kb0,c0 to Σ2 which conjugates Hb0,c0 with σ. 9 2.2 Monodromy Action For (b, c) ∈ HC , then Jb,c = Kb,c is a Cantor set and varies continuously with respect to (b, c). Let K be the space of all points which are bounded in positive and negative time, each associated with their base points:        K = bc yx yx ∈ Kb,c   The restriction of K to the horseshoe locus is a locally trivial bundle of Cantor sets. Given some loop γ : [0, 1] → HC based at (b0, c0), we can follow the points of Jb0,c0 along γ back to the original base point, giving a homeomorphism on Jb0,c0 which commutes with Hb0,c0, and moreover, this homeomorphism depends only on the homotopy type of γ. Because of the canonical identification of Jb0,c0 with Σ2, this induces a natural map ρ : π1(H, (b0, c0)) → Aut(Σ2, σ). The facts that ρ(1) = 1 and ρ(γ1 ◦ γ2) = ρ(γ2) ◦ ρ(γ1) mean that ρ is an antihomomorphism. ρ is called the monodromy action. 10 CHAPTER 3 MONODROMY IN THE HOV REGION 3.1 Inverse Limit Description We know from [HOV94b] that if c M, then there exists some 0 < εc so that for all b ∈ C with 0 < |b| < εc we have (Jb,c, Hb,c) l←i−m−(Jc, fc). Let L be the subset C2 described here. Let π2 : L → (C \ M) be the projection that takes (b, c) to c. For c M, the removal of the two inverse images of the dynamical ray that includes c partitions dynamical space, and there is an isomorphism between Jc and Σ+2 , where one direction (Jc → Σ+2 ) is given by mapping a point to its itinerary relative to this partition. This encoding gives a conjugacy between fc and the shift operator. If c R+, then this isomorphism can be made canonical by assigning the symbol A to the side of the partition that includes the dynamical ray of angle zero and assigning the symbol B to the side of the partition that includes c. For (b, c) ∈ L, we see that the two-symbol coding on Jc induces a two-symbol coding on Jb,c which gives a conjugacy between the maps Hb,c : Jb,c → Jb,c and σ : Σ2 → Σ2. For parameter values in L, the Julia set of the He´non map at γc(t) is the inverse limit system of the quadratic map at π2(γc(t)), so we precisely have the setup described in Appendix A. 3.2 Homotopy of HOV HOV is homeomorphic to the space (C \ {0}) × (C \ D), the product of two spaces which each have homotopy group isomorphic to Z, so Π1(HOV) ≡ Z × Z. 11 Define γb : I → HOV by γb(t) = 1 100 e2πit − 3 and define γc : I → HOV by γc(t) = 1 100 , −3e2πit . γb and γc are both based at p0 and lie in L, and together they generate the fundamental group of HOV. 3.3 Monodromy Action of γb γb projects by π2 down to C as the trivial constant path at a base point. Trivial paths must have trivial monodromy actions, so by Theorem A.1, ρ(γb) = 1. 3.4 Monodromy Action of γc The loop around the Mandelbrot set induces the monodromy action δ on Σ+2 . Hence, by Theorem A.8, ρ(γc) = δ. 12 CHAPTER 4 ORBIT PORTRAITS AND PUZZLES The following chapter will present several definitions which give combinatorial descriptions of one-complex-dimensional quadratic maps. The last section in this chapter is devoted to illustrating these definitions with two examples. 4.1 Formal Orbit Portraits Let us recall a construction from [Mil00]. Definition 4.1. A formal orbit portrait (or abstract orbit portrait) is a finite ordered p-tuple of subsets of the circle P = A1, A2, . . . , Ap such that the following conditions are satisfied: 1. Each A j is a finite subset of R/Z. 2. For each j modulo p, the doubling map t → 2t (mod Z) takes A j bijectively to A j+1 and preserves cyclic ordering of the elements. 3. All of the angles in A1 ∪ · · · ∪ Ap are periodic under angle doubling with common period rp. 4. The sets A1, . . . , Ap are pairwise unlinked. This means that for i j, the two sets Ai and A j can be contained in disjoint, connected subsets of R/Z. Definition 4.2. Let P = A1, . . . , Ap be a formal orbit portrait. For each A j, the connected components of (R/Z) \ A j will be called the complementary arcs of A j. Theorem 4.3 (Milnor). For each A j in a formal orbit portrait P, all but one of the complementary arcs to A j are taken diffeomorphically to the complementary arcs of 13 A j+1 by the angle doubling map. The remaining complementary arc of A j has length greater than 1/2, and its image covers one of the complementary arcs to A j+1 twice and every other complementary arc exactly once. Definition 4.4. Let P = A1, . . . , Ap be a formal orbit portrait. The longest complementary arc for every A j will be called the critical arc for A j. The complementary arc which it covers twice under the doubling map will be called the critical value arc for A j+1 (with the subscripts modulo p). Theorem 4.5 (Milnor). If P is a formal orbit portrait, then among the complementary arcs for the various A j ∈ P, there exists a unique arc IP of shortest length. This shortest arc is the critical value arc for its A j and is contained within all other critical value arcs. This arc is called the characteristic arc for P. 4.2 Actual Orbit Portraits Definition 4.6. An actual orbit portrait O for a one-dimensional quadratic map fc is a repelling or periodic orbit along with the dynamical rays that land on that orbit. Let O be an actual orbit portrait for the map fc and let (z1, . . . , zp) be the associated periodic orbit. Each dynamical ray in an orbit portrait has an associated angle. For i = 1, . . . , n, let Ap be the set of angles of the rays that land at Zi. Then the p-tuple P = A1, . . . , Ap satisfies the four conditions of Definition 4.1 and is hence an orbit portrait. We say that the actual orbit portrait O satisfies the formal orbit portrait P or that fc satisfies the formal orbit portrait. Each actual orbit portrait satisfies exactly one formal orbit portrait, whereas each polynomial fc may satisfy zero, finitely many, or countably many formal orbit portraits. 14 Moreover, Milnor showed precisely where in parameter space a given formal orbit portrait is realized. Theorem 4.7 (Milnor). If P is a formal orbit portrait with characteristic arc IP = [θ−, θ+], then the two parameter rays Rθ− and Rθ+ land at the same parabolic bifurcation point of M. Definition 4.8. The two parameter rays Rθ− and Rθ+ corresponding to the endpoints of the characteristic arc of the abstract orbit portrait P along with their common landing point partition parameter space into two sets: an open set containing 0 and a closed set containing every parameter ray with angle in IP along with a part of M. The latter closed set is called the wake associated with P. There is a 1-1 correspondence between abstract orbit portraits and parameter wakes. Theorem 4.9 (Milnor). fc satisfies P if and only if c ∈ W, where W is the wake associated with P. 4.3 Puzzle Pieces For our purposes, it will be convenient to work with the following definitions of puzzle pieces. Definition 4.10. Given any actual orbit portrait O, removing the associated rays and landing points cuts up the plane into a finite number of open subsets of C. We call these open sets and the finite number of landing points of these rays are called the preliminary puzzle pieces associated with the actual orbit portrait O (or equivalently, associated with the wake W). 15 The preliminary puzzle pieces are some number of singletons and open subsets of the plane. Each ray in an actual orbit portrait maps to another ray in the same portrait. The boundaries of the preliminary puzzle pieces are made up entirely of rays in the portrait. The image of any preliminary puzzle piece under fc is a union of other preliminary puzzle pieces. The mapping is a homeomorphism from each preliminary puzzle piece to a union of other preliminary puzzle pieces, with the sole exception of the preliminary puzzle piece that contains the critical point. This preliminary puzzle piece double covers the preliminary puzzle piece that contains the critical value and singly covers some other preliminary puzzle pieces. We wish to isolate this 2-to-1 behavior. The boundary of the preliminary puzzle piece that contains the critical value is the union of the two rays whose angles are the endpoints of the characteristic arc for the corresponding abstract orbit portrait. We look at the two sets of inverse images of these two rays. One set is already in our actual orbit portrait. The other set is inside the preliminary puzzle piece that contains the critical point. Definition 4.11. Further subdividing the preliminary puzzle piece that contains 0 by the inverse image of the two rays from the characteristic arc gives the puzzle pieces associated with the actual orbit portrait. The two characteristic rays have a common landing point, and this landing point has two distinct inverse images. We also let both of these inverse images be puzzle pieces. Now, the image of each puzzle piece under fc is a union of other puzzle pieces. These maps are all homeomorphisms, with the sole exception of the map from the puzzle piece that contains zero to the puzzle piece which contains the critical value. The puzzle piece that contains the critical point is called the critical puzzle piece (usually denoted Π0) 16 and the puzzle piece that contains the critical value is called the critical value puzzle piece (usually denoted Π1). The mapping from the critical puzzle piece to the critical value puzzle piece is a branched double cover. All of the singleton puzzle pieces are on the repelling or parabolic cycle of the actual orbit portrait, with one exception, and that exception is one of the two inverse images of the singleton puzzle piece that borders the critical value puzzle piece. Both inverse images of this singleton are on the boundary of the critical puzzle piece. The puzzle pieces satisfy the Markov condition. That is to say that if Πi and Π j are puzzle pieces, then either Πi = Π j or Πi ∩ Π j = ∅ and either Π j ⊆ fc(Πi) or fc(Πi) ∩ Π j = ∅. In addition, it is clear that the puzzle pieces form a partition of the Julia set, since the only points of parameter space that are not in the union of the puzzle pieces are the external dynamical rays. Definition 4.12. We can represent the allowed dynamics with an associated directed Markov graph Γ. We define the vertices of Γ to be the puzzle pieces {Πi}. We let there be an arrow Πi → Π j in Γ if and only if fc(Πi) ⊃ Π j. We also sometimes write a double arrow Π0 ⇒ Π1 to signify that fc : Π0 → Π1 is a branched double cover. Lemma 4.13. Let W be a wake with associated abstract orbit portrait P. If c, c ∈ W, then the two Markov graphs coming from the puzzle pieces associated with P for fc and fc are isomorphic. Proof. The puzzles associated with W at c and c have the same sets of rays in each puzzle piece, and they also have the same sets of rays on the boundaries. fc and fc are homeomorphisms an all except for one puzzle piece. The dynamics on the rays is the doubling map on the circle. So where the rays map determines where the puzzle pieces map. 17 The puzzle pieces themselves are not the same throughout the wake, because these are specific subsets of dynamical space, but they do keep the same combinatorics throughout the wake. 4.4 Fattened Puzzle Pieces For every puzzle piece, we will define an associated bounded puzzle piece. Definition 4.14. The bounded puzzle piece Πbi d associated with a given puzzle piece Πi is defined as: Πbi d = Πi ∩ G−1([0, 1)) where G is the associated Green’s function for fc. (The use of the half-open unit interval here is arbitrary. We could have used any interval [0, m) with m > 0 in its place.) For every bounded puzzle, piece, we will define an associated fattened puzzle piece. There are two types of bounded puzzle pieces: singletons and open sets. We will define the associated fattened puzzle pieces for these two types of bounded puzzle pieces separately. For c ∈ int(W), because the periodic cycle associated with O is repelling, we can find small disks D1, . . . , Dp centered at the points of this repelling cycle so that Di+1 is relatively compact in fc(Di). The points on this repelling cycle are on the boundaries of the puzzle pieces. We can make these disks small enough so that their closure does not contain the critical point. (For technical reasons, we also need the disks small enough so that Di ∪ Π j and fc(Di) ∪ Π j are simply connected for every choice of i and j. We also need the images of these disks to have trivial pair-wise intersections and to be contained in the union of the closures of the open bounded puzzle pieces.) 18 Definition 4.15. Define the fattened puzzle piece ∆i associated with a bounded singleton puzzle piece {pi} to be the aforementioned disk Di which contains pi. For the unique singleton that is not in the repelling cycle, its negative is in the repelling cycle, so we use the negative of the fattened puzzle piece around its negative for the fattened puzzle piece of this singleton. Every ray that lands at a point pi in the repelling periodic cycle associated with O must cross ∂Di at some point d. The exterior of the Julia set is open, so there is some distance on either side of d in ∂Di that is also not in the Julia set. Thus, there is also some neighborhood of the angle of the dynamical ray so that the nearby dynamical rays must also cross ∂Di. There are finitely many rays in the actual orbit portrait so there must be some minimum angle ε that all rays can be perturbed and still intersect the same ∂Di. Hence, if Rθ is in the orbit portrait O and lands at pi and |θ − θ | ≤ ε then Rθ must also intersect ∂Di. Definition 4.16. Define the fattened puzzle piece associated with one of the open bounded puzzle pieces as follows: We start with the corresponding bounded puzzle piece. We add on to each bounded puzzle piece the fattened puzzle piece associated with each singleton bounded puzzle piece on its boundary. We also widen each boundary ray by an angle of ε. Each fattened puzzle piece ∆i is associated with one of the original puzzle puzzle pieces Πi. Trivially, we have that Πi ⊂ ∆i. Theorem 4.17. When the arrow Πi → Π j occurs in the Markov graph for O, then ∆ j is relatively compact in fc(∆i). Additionally, if Πi is non-critical, then the map fc is uniformly expanding on all of the points in ∆i that map to ∆ j, with respect to their respective Poincare´ metrics. 19 Proof. When fc maps Πi over Π j, it is easy to see that fc maps each part of the boundary of ∆i outside of the closure of ∆ j. The disks around each point were constructed so that they would map over the closure of the next one. The outer boundary of ∆i will map outside of ∆ j . The dynamics on the rays is the doubling map on angles, so perturbing a ray’s angle of out by ε will perturb the angle of its image out by 2ε. If Πi is non-critical, then Π j is not the critical value puzzle piece, and there are two distinct branches of the inverse of fc on Π j. Let fc−1 be the branch of the inverse that takes Π j inside of Πi. Then fc−1 : ∆ j → fc−1(∆ j) is a holomorphic isomorphism and preserves the Poincare´ metric. Because ∆ j is relatively compact in fc(∆i), then fc−1(∆ j) is relatively compact in ∆i, and hence the inclusion map, ι : fc−1(∆ j) → ∆i is a uniform contraction. Thus, the composition fc−1 : ∆ j → ∆i uniformly contracts Poincare´ metrics. Hence its inverse, fc : fc−1(∆ j) → ∆ j, is uniformly expanding. Corollary 4.18. There exists a metric on an open set containing the closure of the union of the non-critical bounded puzzle pieces for which fc is uniformly expanding. Proof. Let ∆0 be the fattened critical puzzle piece. Let D0 be a small disk centered at zero that does not intersect any non-critical fattened puzzle piece. We can paste together the Poincare´ metrics from every non-critical fattened puzzle piece, using the Poincare´ metric coming from the corresponding fattened puzzle piece when in each bounded puzzle piece. The result gives an expanding metric on the union of the non-critical bounded puzzle pieces. 20 4.5 Itineraries Relative to Orbit Portraits Definition 4.19. The union of the non-critical puzzle pieces has two connected components. Let AWc denote the connected component which contains the landing point of the dynamical ray with angle zero, and let BWc be the connected component that intersects the characteristic dynamical rays. When c ∈ M, then BWc will contain the critical value. Definition 4.20. If fc satisfies an abstract orbit portrait P associated with a wake W and the forward orbit of z never enters the critical puzzle piece of P, then we say that z has a W-itinerary or has an itinerary relative to W. The itinerary it has is the onesided infinite sequence of regions (either AWc or BWc ) that the forward images visit. (We sometimes abbreviate these regions as A and B when c and W are clear from context). 4.6 Kneading Sequences Let us now introduce a construction, called kneading sequences, from [Sch94], which was inspired by [DH82]. 4.6.1 Kneading Sequences of Quadratic Polynomials Choose θ, ϕ ∈ S1. Let the θ-itinerary Iθ(ϕ) of an angle ϕ be a sequence of symbols defined in the following manner: 21  The nth entry of Iθ(ϕ) =   A B B A A B when 2nϕ ∈ θ+1 2 , θ 2 when 2nϕ ∈ θ+1 2 , θ 2 when 2nϕ = θ 2 when 2nϕ = θ+1 2 Schleicher defines the kneading sequence of an angle to be K(θ) = Iθ(θ). Also K+(θ) = limθ θ K(θ ) and K−(θ) = limθ θ K(θ ). Also, K+(θ) is equal to K(θ) with every boundary symbol replaced with the top letter, and K−(θ) is K(θ) with every boundary symbol replaced with the bottom letter. If A is a hyperbolic component of M, then there are two external parameter rays θ− and θ+ landing on its root point. K(θ−) has the same symbols as K(θ+), with the exception that wherever one has the symbol A B in a position, the other has the symbol B A in the same position and vice versa. We define K(A) to be the common symbols fromK(θ+) and K(θ−), except we place a in every position where one has A B and the other has B A . Also we define K−(A) to be the common sequence K−(θ−) = K+(θ+), and we define K+(A) to be the common sequence K−(θ+) = K+(θ−). Definition 4.21. When A is a hyperbolic component of M, we define the characteristic kneading sequence of A to be K+(A). Theorem 4.22 (Schleicher). Let A be a hyperbolic component with angles θA− and θA+ landing at its root and B be a hyperbolic component with angles θB− and θB+ landing at its root with A ≺ B (or equivalently 0 < θA− < θB− < θB+ < θA+ < 1), and ϕ in either (θA− , θB−) or (θB+, θA+ ). If there is no hyperbolic component of period k or less between A and B, then the kth term in the sequences K+(A), K−(B), K(ϕ) are all identical. The kneading sequence of an angle can only change at the ith position when moving across a point of the circle that is periodic with period i. In fact, it must change. And 22 when the angle is on the periodic cycle of period i, the ith term must be either A B or B A . So, the first k terms in the itineraries of points of the circle are constant on intervals of the circle where there are no periodic cycles with period less than or equal to k. Definition 4.23. We say a hyperbolic component B is conspicuous to a hyperbolic component A if A ≺ B, B has a period no greater than that of A and there are no hyperbolic components of period lower than that of B between the two in the ≺ ordering. Note that a wake is always conspicuous to itself, and for a given wake, there can only be finitely many wakes conspicuous to it. Also, as a relation, conspicuousness is not transitive or commutative. For readers familiar with the visibility relationshio, conspicuousness is similar to visibility, though they are distinct. Compare with [Sch94]. The author is indebted to Dierk Schleicher for the idea behind the proof of the following theorem: Theorem 4.24. Let W be the wake of a hyperbolic component A. There are finitely many hyperbolic components conspicuous to A. Let these be A1, . . . , Ar. Choose any parameter ray Rθ inside of W that is not one of the finitely many rays that land on the root points of the components conspicuous to A. Then there is some Ai so that K(θ) and K+(Ai) have a common prefix of length m, where m is the period of Ai. Proof. We use induction on the ordering of wakes. Let n be the period of A. If the only component conspicuous to A is itself then then there are no periodic angles in W with period less than the period of A. Hence, the first n terms of the kneading sequences of angles must be constant inside of W. Hence every angle in W has a prefix of the characteristic kneading sequence of A. Alternately, if r > 1, then either Rθ is contained only in A and no other conspicuous 23 component, or Rθ is contained in some conspicuous component other than A. If the latter, we induct on this component. If the former, then let θA− and θA+ be the two parameter rays that land on the root point of A. For any period k ≤ n, there must be an even number of angles between θA− and θ of period k under doubling, for if there was an odd number, then there would have to be at least one wake of period k that contains Rθ and is contained in W. And Rθ would have to be contained in some wake conspicuous to W. Since there are an even number of periodic angles of period k between θA− and θ, then the kth term in the kneading sequence must flip an even number of times between these two angles, so the kth terms of the sequences K+(A) and K(θ) must be the same. This is true for every k ≤ n, so K+(A) and K(θ) have the same initial length-n string. Note also that if θ is one of the rays that land on the root points of the components then K(θ) is not, strictly speaking, an AB coding, but both K+(θ) and K−(θ) satisfy the consequent of the theorem statement (except for the two rays landing on A, and in this case, exactly one of K−(θ) and K+(θ) do). 4.6.2 Kneading Sequences of Orbit Portraits We will define a characteristic kneading sequence for an orbit portrait and the associated wake. Let P be an abstract orbit portrait of period n with associated wake W. Choose some c ∈ W. Then fc has an actual orbit portrait O that satisfies P. Definition 4.25. The characteristic kneading sequence, K(W), of W will be a word in A’s and B’s of length n and will be constructed as follows: Start at the point of the 24 periodic orbit in O that borders on Π1, and list off the two-symbol itinerary of this point relative to W up to, but not including the point on the orbit that borders Π0. The nth symbol will be the opposite of the two symbol coding of this point that borders Π0. This definition is somewhat counter-intuitive. The intuition for this choice is that the characteristic kneading sequence is the initial segment of the kneading sequences of polynomials that are “just-beyond” the hyperbolic component at the root of W. We see this in the following theorem: Theorem 4.26. The characteristic kneading sequence for a wake of period n with hyperbolic component A at its root is precisely the first n terms in K+(A). Proof. Let P be the abstract orbit portrait associated with W. Let θ1− be the angle of the smaller of the two rays that borders the critical value puzzle piece. Let θ0− be the inverse image of θ1− that is in P. Choose ε to be a small positive num- ber. Then K(θ1− + ε) is the itinerary of θ1 + ε under the partition of the circle (θ0− + ε 2 , θ0− + ε 2 + 1 2 ), (θ0− + ε 2 + 1 2 , θ0− + ε 2 ) . If ε is small enough, then the first n−1 terms in this itinerary will be identical to the W-itinerary of the point of the periodic orbit that borders Π1. The difference comes at the nth term. 2n−1(θ1− + ε) = θ0− + 2n−1ε, which (when ε is small enough) is on the opposite side of the partition as θ0−, which is the angle of the external parameter ray that lands on the point of the orbit bordering Π0. Hence the nth term of K+(A) is the opposite of the W-coding of the point of the periodic orbit that borders Π0. It is worth noting that if A is the hyperbolic component at the root of the wake W, then K+(A) is not the same as K(W). K+(A) is a one-sided infinite sequence on two symbols. K(W) is a finite string of n symbols (where n is the period of A and W) and is just the first n symbols of K+(A). 25 Corollary 4.27. Let Rθ be a parameter ray that lands in the interior of a wake W. Then either there is some wake W conspicuous to W so that K(θ) begins with K(W ) or Rθ is one of the boundary rays of a wake conspicuous to W. If the latter, then both K−(θ) and K+(θ) begin with the characteristic kneading sequences of (different) wakes. Proof. Consequence of Theorems 4.24 and 4.26. Thus, we see that the characteristic kneading sequence for a wake describes the initial segments of the kneading sequences for the group of polynomials that come after the hyperbolic region at the base of the wake, but before smaller wakes of lower period. Sometimes we know the initial segments of the kneading sequence for all the subsequent polynomials (when there are no smaller conspicuous wakes). Other times we only know the initial segments of kneading sequences of polynomials in a wake that are not in one of the wakes conspicuous to that wake. 4.7 Examples The following two examples will illustrate the definitions in this chapter. 4.7.1 The Airplane The following ordered triple of two-element sets is a formal orbit portrait: Pair = ({3/7, 4/7} , {6/7, 1/7} , {2/7, 5/7}) Here, A1 = {3/7, 4/7}, A2 = {6/7, 1/7}, A3 = {2/7, 5/7}, and it is easily verified that Pair satisfies the four conditions of Definition 4.1. 26 The complementary arcs of {3/7, 4/7} are (3/7, 4/7) and (4/7, 3/7). The complementary arcs of {6/7, 1/7} are (6/7, 1/7) and (1/7, 6/7). The complementary arcs of {2/7, 5/7} are (2/7, 5/7) and (5/7, 2/7). Among these, the critical arcs are (4/7, 3/7), (1/7, 6/7), and (5/7, 2/7). The critical value arcs are (3/7, 4/7), (1/7, 6/7), and (2/7, 5/7). The characteristic arc of Pair is (3/7, 4/7). Figure 4.1: M with 3/7 and 4/7 parameter rays. As shown in Figure 4.1, the two rays of angles 3/7 and 4/7 land at the same point of the Mandelbrot set. Let Wair be the wake associated with Pair. The two parameter rays 3/7 and 4/7 form the boundary of Wair. Wair is the region of parameter space to 27 the left of these two boundary rays in Figure 4.1. Figure 4.2: Airplane polynomial with actual orbit portrait Oair Every polynomial in Wair satisfies the formal orbit portrait Pair. There is one such polynomial at approximately c ≈ −1.75 called the airplane polynomial. The airplane polynomial is characterized as the unique real quadratic polynomial with a super-attracting period-three cycle. There is an actual orbit portrait Oair for the airplane polynomial which satsifies Pair. Figure 4.2 shows the Julia set of the airplane along with Oair. Figure 4.3 shows the nine puzzle pieces associated with Pair. The five open puzzle 28 Figure 4.3: Oair and the Julia set of airplane polynomial pieces are Π0, Π1, Π2, Π3, and Π4. The four singleton puzzle pieces are Π5, Π6, Π7, and Π8, and are represented in Figure 4.3 by green dots along the real axis. Let Γair be the Markov graph of the allowable transitions between puzzle pieces of Pair. Γair is illustrated in Figure 4.4. Let Aair be the hyperbolic component of M which contains the airplane polynomial. Then for Aair, θ− = 3/7 and θ+ = 4/7. 29 10 2 34 57 8 6 Figure 4.4: Γair K(θ−) = BA A B K+(θ−) = BAA K−(θ−) = BAB K(θ+) = BA B A K+(θ+) = BAB K−(θ+) = BAA K(Aair) = BA K+(Aair) = BAA 30 K−(Aair) = BAB K(Wair) = BAA The period of Wair is 3. There are no wakes contained within Wair of lower period. Thus, the only wake conspicuous to Wair is itself. The fact that no other wakes are conspicuous to Wair implies that every angle between 3/7 and 4/7 has a kneading sequence which begins with the three symbols BAA. Also implied is the fact that the kneading sequence of every hyperbolic component C for which Aair ≺ C begins with the string BAA. Additionally, every wake contained in Wair has a characteristic kneading sequence which begins with BAA. 4.7.2 BABB The following ordered quintuple of two-element sets is a formal orbit portrait: PBABB = 13 , 18 , 26 5 , 10 , 21 , 20 , 11 , 22 , 9 31 31 31 31 31 31 31 31 31 31 The characteristic arc of PBABB is (13/31, 18/31). The two parameter rays at angles 13/31 and 18/31 land on the same point of the Mandelbrot set, as is illustrated in Figure 4.5. Let WBABB be the wake associated with PBABB and illustrated in Figure 4.5. Every polynomial in WBABB satisfies PBABB. Define ABABB to be the hyperbolic component of M at the base of WBABB. Let fBABB be the polynomial at the center of ABABB. fBABB has a super-attracting cycle of period 5 and satisfies PBABB. There is an actual orbit portrait OBABB for fBABB associated with PBABB. Figure 4.6 illustrates OBABB along with the Julia set of fBABB. 31 Figure 4.5: Mandelbrot set with 13/31 and 18/31 parameter rays Figure 4.7 illustrates the puzzle piece decomposition of dynamical space associated with PBABB for the polynomial fBABB. The open puzzle pieces are Π0, Π1, Π2, Π3, Π4, Π5, and Π6. The singleton puzzle pieces are all on the real axis and are marked with green dots. These are Π7, Π8, Π9, Π10, Π11, and Π12. Let ΓBABB be the Markov graph describing the possible transitions between these puzzle pieces. Figure 4.8 shows ΓBABB. For the hyperbolic component ABABB, θ− = 13/31 and θ+ = 18/31. The following 32 Figure 4.6: OBABB and the Julia set of fBABB may be computed: K(θ−) = BABB A B K+(θ−) = BABBA K−(θ−) = BABBB K(θ+) = BABB B A K+(θ+) = BABBB K−(θ+) = BABBA 33 Figure 4.7: Puzzle pieces for fBABB associated with PBABB K(ABABB) = BABB K+(ABABB) = BABBA K−(ABABB) = BABBB K(WBABB) = BABBA Wair is contained in WBABB and there are no other wakes W of period lower than 3 (the period of Wair) for which WBABB ≺ W ≺ Wair. Also, the period of Wair is less than the period of WBABB. Thus, Wair is conspicuous to WBABB. 34 106 2 8 12 7 3 4 5 10 9 Figure 4.8: ΓBABB 11 Let WBAA be the period 4 wake whose boundary is the union of the two parameter rays at angles 7/15 and 8/15. WBAA is contained in WBABB. Also, the period of WBAA is 4, which is less than the period of WBABB, which is 5. However, WBAA is not conspicuous to WBABB, because WBABB ≺ Wair ≺ WBAA, and Wair has a smaller period than WBAA. Every wake is conspicuous to itself, so the only two wakes conspicuous to WBABB are itself and Wair. 35 CHAPTER 5 XCW 5.1 Defining XWc Definition 5.1. Choose any parameter wake W with associated abstract orbit portrait P. Choose any c ∈ int (W). Then fc has an actual orbit portrait O with a repelling periodic cycle that satisfies P. O has a critical puzzle piece Π0. Let XWc = x ∈ Jc (∀n ∈ N0) fc◦n(x) Π0 XWc is the set of all points in the Julia set that do not visit the critical puzzle piece. It is clear that XWc is forward-invariant under fc. XWc is not backwards-invariant, because points in the critical value puzzle piece have inverse images in the critical puzzle piece. These are the only points of XWc that do not have two inverse images in XWc . It is also clear that points in XWc have a well defined two symbol W-itinerary. Lemma 5.2. If W ⊆ W are wakes and c ∈ int (W ), then XWc ⊆ XWc Proof. The critical value puzzle piece of W is contained in the critical value puzzle piece of W, so we also have containment of their respective critical puzzle pieces. Theorem 5.3. fc is uniformly expanding on XWc . Proof. XWc is contained in the union of the non-critical bounded puzzle pieces. Corollary 5.4. XWc ⊂ Jc and every point of XWc is accessible. 36 Proof. The statement is trivial for the singleton puzzle pieces since they are the landing points of rays of the orbit portrait. We have expansion on a neighborhood of every other point in XWc . Corollary 5.5. A point of XWc is determined by its one-sided two-symbol coding relative to W. Proof. We have expansion on the union of all of the non-critical puzzle pieces. fc restricted to the union of the puzzle pieces on either side of the critical puzzle piece is a homeomorphism. So if two points are distinct, then they must eventually map to different sides of the critical puzzle piece. Theorem 5.6. If c, c ∈ int (W), then the W-itineraries realized by points in (respectively) XWc and XWc are identical subsets of Σ+2 . Proof. Suppose that the itinerary ε is realized at XWc by a point x, which is the landing point of a ray Rθ. Note that ε = K(θ). Under the doubling map, θ never enters the inverse image under the doubling map of the characteristic arc of the wake W. Hence, because of the expansion on the non-critical puzzle pieces for fc , the dynamical ray of angle θ for the map fc must land at some point x ∈ XWc . The W-itinerary of x under the action of fc must be K(θ), the same as for x. Corollary 5.7. There exists some Σ+W ⊂ Σ+2 , depending only on W, so that for every c ∈ int (W), taking the W-itinerary gives an isomorphism from XcW, fc to Σ+W, σ . Proof. If two points in XWc have the same W-itinerary, then by the expansion on AWc and BWc , they must be the same point. Thus we have an injective map from XWc to Σ+2 . Let Σ+W be the image of this map. By Theorem 5.6, Σ+W is independent of our choice of c. We have a bijection from XWc to Σ+W that conjugates fc with the shift. 37 5.2 Multi-Itineraries The fattened puzzle pieces cover XWc , but they do not partition it. The fattened puzzle pieces have non-trival intersections. Definition 5.8. A multi-itinerary of a point x in dynamical space with respect to a map f relative to a cover C = {∆i} is sequence of elements of the power set of C, (C0, C1, C2, . . .) such that f n(x) ∈ ∆i if and only if ∆i ∈ Cn. A multi-itinerary is a natural extension of the concept of an itinerary to situations where we do not have an explicit partition of dynamical space. 1 The multi-itinerary simply keeps track of all of the regions that the point lands in under iteration, even when it lands in more than one. The puzzle pieces have the property that the image of each puzzle piece contains all of the other puzzle pieces to which there is an arrow from that puzzle piece in Γ. This property is not stable under a small C1 perturbation, which is why we must use fattened puzzle pieces. The disadvantage of using fattened puzzle pieces is that we lose the partition of dynamical space. This introduces some difficulties which we will have to carefully deal with. The non-critical fattened puzzle pieces cover XWc , so points in XWc have a multiitinerary relative to the non-critical fattened puzzle pieces. Theorem 5.9. The multi-itinerary of a point x ∈ XWc relative to the fattened puzzle pieces determines that point’s itinerary relative to the non-fattened puzzle pieces. 1Given sets Cα, indexed by α ∈ ℵ, the multi-itinerary is the itinerary relative to the partition:   α∈ℵ Cα if I(α) = 1 (Cα)c if I(α) = 0  I ∈ 2ℵ  38 Proof. Given a multi-itinerary (C0, C1, . . .) of x, we will give an algorithm for how to construct x’s itinerary with respect to unfattened puzzle pieces which uses only the multi-itinerary. Let xn = fc◦n(x). Let Ci be the first term in the itinerary that contains more than one fat puzzle piece. Then the point is inside one of the small disk-like puzzle pieces around a singleton puzzle piece. Then one of two situations occurs. Either every term in the itinerary subsequent to Ci includes multiple puzzle pieces or there is some minimum j > i so that C j does not include any small disk puzzle piece. If the former, then by the expansion, we know that xi is a point in the periodic cycle, and Ci and every subsequent term can be replaced with the unfattened singleton puzzle piece that the corresponding disks contain. If the latter, then because the small disk puzzle pieces are small enough, we know that when a point leaves the region where it has a multi-itinerary, then on the next iterate, it must be in one of the unfattened puzzle pieces adjacent to the next small disk puzzle piece. Because the map is a local homeomorphism near the points of the periodic cycle, then this lets us resolve which unfattened puzzle piece from C j−1 that x j−1 is in. Note that we do not have to worry about x j landing in multiple fattened puzzle pieces that are not small disks, because these regions lie entirely outside of the Julia set. This procedure can be applied iteratively to resolve all of the terms between Ci and C j, and then this process can be iteratively applied (possibly infinitely many times) to resolve the rest of the terms. Thus, we can determine the coding of x with respect to the puzzle pieces (which do form a Markov partition) from the coding of x with respect to the fattened puzzle pieces (which do not). Moreover, everything in the proof of Theorem 5.9 holds for bi-itineraries of points 39 in the inverse limit system of XcW, fc as well as for small enough C1 perturbations of fc or its inverse limit. 5.3 Adaptation to Γ Definition 5.10. Let U be a simply connected Riemann surface homeomorphic to the disk, and U a relatively compact open subset. Define the size of U in U to be 1/M , where M is the largest modulus of an annulus separating U from the boundary of U . Definition 5.11. Sullivan defines a p-telescope to be a sequence of topological disks (W0, W1, . . .) such that Wi+1 is relatively compact in p(Wi). Theorem 5.12 (Sullivan). If (W0, W1, . . .) is a p-telescope, and Wi+1 is of uniformly bounded size in p(Wi), then the intersection ∩∞i=0 p◦−i(Wi) is a single point. Proof. Standard application of Schwarz-Pick lemma along with properties of moduli show that the Poincare´ metrics in these disks are uniformly expanding under fc. We wish to have an isomorphism between points of XWc and paths in the graph Γ. The obvious map from XWc to paths in Γ takes a point to its itinerary relative to the bounded puzzle pieces. This map is well-defined and is injective. The obvious map going the other direction takes an itinerary of non-critical puzzle puzzle pieces, fattens them, and then uses telescopes to identify a unique point of XWc . This action is welldefined, but unfortunately, it is not always injective. In the case of primitive orbit portraits, there are always two itineraries that map to any pre-image of the periodic cycle. The reason for this is that when W is primitive, there is a path in Γ of non-critical open puzzle pieces where successive pieces neighbor 40 the successive points on the repelling periodic cycle associated with W. Thus, the fattened puzzle pieces will include the successive points of the periodic cycle, and the telescope will identify that point on the periodic cycle. To get our desired isomorphism, we must exclude such paths. Definition 5.13. Call a path (Πi0, Πi1, Πi2, . . .) in Γ degenerate if every Πin is an open puzzle piece and there exists some x in the periodic cycle of the orbit portrait so that fcn(x) ∈ ∂Πin for every n ∈ N0. We call such paths degenerate because the point they identify in XWc using telescopes is not in the first puzzle piece of the itinerary. If W is satellite, there are no degenerate paths. If W is primitive, there is a single cycle of degenerate paths with the same period as W. Definition 5.14. We say that a one-sided or two-sided infinite path in Γ is adapted to Γ if it does not include the critical puzzle piece and no right-infinite tail of the path is degenerate. Theorem 5.15. There is an isomorphism between XWc and paths adapted to Γ. Proof. The map from XWc to paths is given by itineraries. An itinerary of a point can never be degenerate. Because of expansion on XWc , this map is injective. The map from paths to XWc is given by first fattening the puzzle pieces and then taking a telescope. The strong expansion on the puzzle pieces ensure that this construction is well-defined. Now we will show that this procedure is injective. Suppose that (Πi0, Πi1, Πi2, . . .) and (Π j0, Π j1, Π j2, . . .) are two different non-critical paths in Γ yield the same point x under the action of fattening and taking telescopes. Let 41 xk = fc◦k(x). xk ∈ ∆ik ∩ ∆ jk for all k. There is some Πin Π jn. Fattened puzzle pieces overlap on the disks around the points in the orbit portrait, so at least one of Πin and Π jn is an open puzzle piece that borders some point zn on the periodic orbit. Without loss of generality, let Πin be the open puzzle piece. Let zk+n = fc◦k(zn). Because fc is a homeomorphism when restricted to either side of the critical puzzle piece, then Πim Π jm, Πim is open, and zm ∈ ∆im ∩ ∆ jm for all m ≥ n. Hence, we see that (Πi0, Πi1, Πi2, . . .) has a degenerate tail. So whenever two itineraries are associated with the same point in XWc , then one must have a degenerate tail. Thus the map that associates itineraries adapted to Γ with points is also injective. It is also clear that these two actions are inverses of each other. Definition 5.16. We say that a one-sided multi-itinerary is adapted to Γ if it is the multiitinerary of of some point of XWc relative to the fattened puzzle pieces. The action that takes a point of XWc to its multi-itinerary adapted to Γ is surjective because of how adaptation to Γ is defined for multi-itineraries and is injective because of expansion on XWc . Definition 5.17. We say that a two-sided multi-itinerary is adapted to Γ if every rightinfinite tail of it is adapted to Γ. 5.4 Relations Between Points and Itineraries The following commutative diagram illustrates isomorphisms between points in dynamical space, itineraries relative to various partitions, and multi-itineraries. All of these maps commute with the shift operator or fc (whichever is the appropriate operator on that space). 42 }{{{{{{{{f{{{{{{{{{{= Σ+W `@@@@@@@g@@@@@@@@@ XWc d  oPPPPPPPPPPPPPPPcPaPPPPPPPPPPItPipnPiee' rcaersieasdoaO fpbftaetdptuozΓzle Multi-itinerary of fat puzzle pieces adapted to Γ e / Itinerary of puzzle pieces adapted to Γ a) The puzzle pieces partition XWc , so every point has an itinerary adapted to Γ. b) The correspondence between puzzle pieces and their fattened counterparts gives a trivial correspondence between bi-infinite sequences adapted to Γ. c) An itinerary of fattened puzzle pieces adapted to Γ gives a unique point using telescopes. d) Points of XWc have multi-itineraries with respect to the fattened puzzle pieces. e) Theorem 5.9 gives an algorithm for determining a point’s itinerary given only its multi-itinerary relative to the fattened puzzle pieces. f) XWc → Σ+W is given by the definition of a W-itinerary. Σ+W → XWc is Corol- lary 5.5. g) Non-critical puzzle pieces are always on one side or the other of the critical puzzle piece. 43 5.5 Continuity of XWc Theorem 5.18. XWc varies continuously with c. Proof. By Corollary 5.7, every point of XWc is identified with a unique W-itinerary and the set of realizable W-itineraries is independent of the choice of c ∈ int (W). Define ΦWc,c (x) to be the point whose W-itinerary under fc is the same as that of x under fc. What we need to show that the function ΦWc,c (x) : W → C is a continuous function of c as x, c, and W are held constant. Also note that ΦWc,c : XWc → XWc is the identity. We will show that ΦWc,· (x) is continuous on a small open domain around c. x ∈ XWc has a W-itinerary: (P0, P1, . . .). Let fc ,Pi be the restriction of fc to the union of either the A- or B-side fattened puzzle pieces, depending on the symbol Pi. Then ∞ ΦWc,c (x) = fc−,1P0 · · · fc−,1Pn (C) n=1 For i, j ∈ N0 with i ≥ j, define: zi, j = f −1 c ,Pi− j ··· f −1 c ,Pi−1 fci(x) zi, j is the point you get when you map x forward i times by fc, and then pull back j times by fc , each time taking the appropriate branch of the inverse. Once we fix c, x, and W (as we have), then zi, j is a function of c . If limn→∞ zn,n exists, then this limit must equal ΦWc,c (x), because it has the appropriate W-itinerary under fc . We will prove convergence by showing that successive distances between points on the sequence (z0,0, z1,1, z2,2, . . .) are bounded geometrically with a uniform contraction constant on a small neighborhood of c. In addition, we will show that ΦWc,· (x) is continuous by showing that the initial 44 constant of the geometric series is bounded by a constant multiple of |c − c | in another open set around c. For any zi,0 ∈ AWc ∪ BcW, then the mean-value theorem applied to the square-root function gives: zi+1,1 − zi,0 = fc−,1Pi ( fc(zi,0)) − zi,0 = ± z2i,0 + c − c − ± z2i,0 1 ≤ max √ · |c − c| 2 xx∈ z2i,0+t(c−c )|t∈I 1 ≤ |c − c| 2 z2i,0 − |c − c| ≤ k0 |c − c| Obviously here, we must restrict c to a domain U around c such that U ⊂ int(W) and small enough so that even for w ∈ U, then |w − c| is smaller in absolute value than the square of any point in either the AWc or BWc regions. And we let  k0 = sup y∈AWc ∪BWc  2 c ∈U  y2 1 − |c − c|  Hence zi,0 − zi+1,1 ≤ k0 |c − c| for all i ∈ N0 and c ∈ U. Let ∆mi be the fattened puzzle piece for fc whose corresponding non-fattened piece for fc contains zi,0, and let di(·, ·) be the Poincare´ metric on this piece. The periodic cycle of the orbit portrait associated with W is well-defined and repelling in int (W), so it and the rays that land on it move continuously. Hence, the nonfattened and fattened puzzle pieces associated with W move continuously for c ∈ W. Let V be an open region in parameter space around c whose closure is in int(W) and 45 such that whenever c ∈ V, then the closures of the bounded puzzle pieces for fc are contained in the fattened puzzle pieces for fc and the closures of the bounded puzzle pieces for fc are contained in the fattened puzzle pieces for fc. Associate with every puzzle piece Πi at every parameter value c ∈ V an open set S i that moves continuously with c , contains the closure of the ith puzzle pieces for fc and fc , and is contained within the corresponding fattened puzzle pieces for both fc and fc . We apply a constant multiple to the Poincare´ metrics on each fattened puzzle piece so that the resulting metrics are strictly greater than the Euclidean metrics. For every c ∈ V, on every S i, the Poincare´ metrics on the corresponding fattened puzzle pieces are equivalent to the Euclidian metric. There is some some k2 > 1 so that for all c ∈ V, all S i at c , and all x, y ∈ S i, then |x − y| < di(x, y) < k2 |x − y| We have uniform expansion on the map fc : ∆i → ∆ j whenever Πi → Π j is in Γ. There are finitely many arrows in Γ, and this expansion depends continuously on c , so there must be some k1 < 1 so that k1 · dn+1(x, y) > dn fc−,1Pn(x), fc−,1Pn(y) whenever c ∈ W and x, y ∈ ∆mn+1. Fix c ∈ V for the following discussion. di zi,0, zi+1,1 ≤ k2 zi,0 − zi+1,1 ≤ k2k0 |c − c| By induction on the contraction of fc−,1Pn, di− j zi, j, zi+1, j+1 ≤ k1jk2k0 |c − c| Setting i = j yields d0 zi,i, zi+1,i+1 ≤ k1i k2k0 |c − c| 46 We see that the sequence (z0,0, z1,1, z2,2, . . .) is Cauchy and converges. By geometric summation and repeated application of the triangle inequality, x − ΦWc,c (x) = z0,0 − lim zn,n n→∞ ≤ d0 z0,0, lim zi,i i→∞ ≤ k2k0 |c − c| 1 − k1 Since this is true in an open neighborhood W around every c ∈ int(W), then ΦWc,c (x) is locally Lipshitz and hence continuous in c . 5.6 W-itineraries of XWc Theorem 5.19. Points in XWc whose W-itinerary contains K(W) must lie in Π1. Also, K(W) can only appear as the initial segment of W-itineraries of points in XWc . Proof. Let [θ−, θ+] be the characteristic arc of W. Let n be the period of W. Suppose that z ∈ XWc contains the characteristic kneading sequence of W, K(W) = (χ0, . . . , χn−1) as a substring. Then let z0 be the iterate of z such that K(W) is the initial segment of the itinerary‘ of z0. Label the further iterates of z0 by zi+1 = fc(zi). Let p0 through pn−1 be the points of the repelling or parabolic cycle of W with the labeling such that p0 is the landing point of Rθ− and Rθ+. (There are not necessarily n distinct points in this cycle, but we only need their cyclical order for the following, not their distinctness.) For any particular p j, the k dynamical rays that land on it partition the dynamical plane into sectors based at p j. The essential fact is that sectors based at p j map to sectors based at p j+1. All but one of these sectors maps homeomorphically to the next. The critical sector (the sector that contains the critical puzzle piece) is the odd one out. The critical puzzle piece maps in a branched 2-to-1 fashion to the critical 47 value puzzle piece. Removal of the critical puzzle piece sometimes splits the critical sector into two parts, sometimes not. In either case, the connected components of the remainder map homeomorphically to their images. The dynamical rays Rθ− and Rθ+ together land on p0 and cut off the critical value puzzle piece of W. Then R2n−1θ− and R2n−1θ+ land on pn−1, and together make up one edge of the boundary of the critical value puzzle piece. Also notice that 2nθ− = θ− and 2nθ+ = θ+. The sector based at pn−1 bounded by R2n−1θ− and R2n−1θ+ contains the critical puzzle piece as well as the other half of dynamical space which is across the critical puzzle piece from pn. This sector must therefore contain all of either AWc or BWc , whichever has the opposite coding as pn−1 itself. Our supposition that the point z0 has a kneading sequence which is the characteristic kneading sequence of the wake W therefore ensures that zn−1 is in the sector based at pn bounded by R2n−1θ− and R .2n−1θ+ Now, assume that zi+1 is in the sector bounded by R2i+1θ− and R2i+1θ+ based at pi+1. If that sector does not contain the critical point, then its inverse image has two connected components, each of which maps homeomorphically to the sector. One of these connected components is in AWc and the other is in BWc . If that sector does contain the critical point, then it contains Π1, and its inverse image is connected and contains the critical puzzle piece. After removal of the critical puzzle piece, there are two remaining connected regions, each of which mapping homeomorphically to the original sector minus the critical value puzzle piece. Again, one of these connected components is in AWc and one of them is in BWc . As mentioned earlier, points of XWc can never have any iterate (including themselves) in Π0, so zi+1 has two distinct inverse images, one in AWc and one in BWc . One of these is zi. Which one is zi depends on the ith entry in the itinerary of z0, and hence on the ith entry of the characteristic kneading sequence of W. Thus, zi and pi 48 are in the same one of AWc and BWc . The map fc|AWc : AWc → C and fc|BWc : BWc → C are both homeomorphisms onto their images. So pulling back R ,2i+1θ− R ,2i+1θ+ pi+1 and zi+1 by the appropriate one of f |A or f |B (whichever one pi and zi are in) must give the appropriate containment relationship: that zi is contained in the sector based at pi bounded by R2iθ− and R2iθ+. By induction, z0 is contained in the critical value puzzle piece. Since points in the critical value puzzle piece cannot have inverse images, then we see that z = z0. Hence, z lies in the critical value puzzle piece and K(W) is the initial segment of the W-itinerary of z. Theorem 5.20. Points in XWc whose W-itinerary contains the substring K(W ) for any W conspicuous to W must lie in Π1. Also, these substrings can only appear as the initial segment of itineraries of points in XWc . Proof. Suppose the itinerary of x contains the characteristic kneading sequence of some W ⊂ W. Choose some c ∈ int (W ) ⊆ W. XcW can be followed continuously to XcW, where x ∈ XWc is followed to some x ∈ XWc . Because the two external rays and the point of the periodic orbit that together bound the critical value puzzle piece move continuously, then either both x and x are in their respective critical value puzzle pieces associated with the wake W for fc and fc or neither are. Similary, the rays that bound the critical puzzle piece move continuously, so x and x have the same W-itinerary. x contains in its W -itinerary the characteristic kneading sequence of W , so by Theorem 5.19, then x is in the critical value puzzle piece associated with W , and K(W ) may only appear at the beginning of the W -itinerary of x . Since W ⊆ W, then we have containment of their respective critical value puzzle pieces, and we see that x must be in the critical value puzzle piece of W for fc . Also, whenever both are defined, then the W-itinerary of a point is equal to the W -itinerary of the point. Hence 49 x is in the critical value puzzle piece for W for the polynomial fc and K(W ) can only appear at the beginning of its W-itinerary. Theorem 5.21. Choose x ∈ XWc . If x is in the critical value puzzle piece, then some initial string of the two-symbol itinerary of x relative to W is equal to the characteristic kneading sequence of some conspicuous wake W ⊆ W. Proof. Suppose x ∈ XWc and x ∈ Π1. x is accessible, so there is some dynamical ray Rθ that lands on x. The two inverse images Rθ/2 and Rθ/2+1/2 both lie in the critical puzzle piece. The forward images of x never enter the critical puzzle piece, so the W-itinerary of x must be the same as K(θ). Since x never again enters the critical value puzzle piece, then x and θ cannot be periodic. Thus, the parameter ray Rθ cannot lie on the boundary of some wake smaller than W. By Corollary 4.27, the W-itinerary of x must begin with K(W ) for some conspicuous wake W of W. Corollary 5.22. Choose x ∈ XWc . The W-itinerary of x contains the characteristic kneading sequence of some wake conspicuous to W if and only if x is in the critical value puzzle piece. Additionally, the characteristic kneading sequence of the wake conspicuous to W may only appear as the initial segment of the itinerary of any point of XWc . Proof. The “if” direction is given by Theorem 5.21. The “only if” direction is given by Theorem 5.20. Corollary 5.23. The set of one-sided two-symbol W-itineraries of points in fc(XWc ) is a one-sided subshift of finite type where the disallowed words are the characteristic kneading sequences of wakes conspicuous to W. Proof. fc(XWc ) can be characterized as the set of points which do not enter the critical value puzzle piece under iteration. Then by Corollary 5.22, the W-itineraries realized 50 by fc(XWc ) are all of the one-sided sequences on two symbols that do no have any characteristic kneading sequence for any wake conspicuous to W. l←i−m−(XWc , fc) naturally inherits a two-sided W-itinerary from the one-dimensional system. Corollary 5.24. The set of two-sided two-symbol W-itineraries realized by l←i−m−(XWc , fc) is a two-sided subshift of finite type where the disallowed words are the characteristic kneading sequences of wakes conspicuous to W. Proof. Note that every point in XWc has two inverse images in XWc with the exception of points in the critical value puzzle piece. The inverse images of these points are in the critical puzzle piece and hence not in XWc . Hence l←i−m−(XWc , fc) is made up of orbits which never visit the critical value puzzle piece, or equivalently, those that do not have any characteristic kneading sequence of any wake conspicuous to W. 51 CHAPTER 6 XWB,C 6.1 Crossed Mappings Let us recall from [HOV94b] the definition of a crossed mapping: Definition 6.1. Let B1 = U1 × V1 and B2 = U2 × V2. Let pr1 : Bi → Ui be the projection to the first co-ordinate and let pr2 : Bi → Vi be projection to the second co-ordinate. A crossed mapping from B1 to B2 is a triple (W1, W2, f ), where 1. W1 ⊆ U1 × V1 where U1 ⊂ U1 is a relatively compact open subset, 2. W2 ⊆ U2 × V2 where V2 ⊂ V2 is a relatively compact open subset, 3. f : W1 → W2 is a holomorphic isomorphism, such that for all y ∈ V1, the mapping pr1 ◦ f |W1∩(U1×y) : W1 ∩ (U1 × {y}) → U2 is proper, and the mapping pr2 ◦ f −1|W2∩({x}×V2) : W2 ∩ ({x} × V2) → V1 is proper. When W1 and W2 can be determined from context, we write f : B1 →− B2. × Each Bi has a Kobayashi metric, and when Ui and Vi are disks, the Kobayashi metric on Bi has the simple form of the product of the Poincare´ metrics on Ui and Vi, which will be denoted |·|Ui and |·|Vi, respectively. Definition 6.2. An analytic curve in B = U × V is called horizontal-like if at every point (x, y) ∈ U × V, the tangent vector at that point (η, ν) ∈ T(x,y)B satisfies |(x, η)|U ≥ |(y, ν)|V. 52 Definition 6.3. An analytic curve in B = U × V is called vertical-like if at every point (x, y) ∈ U × V, the tangent vector at that point (η, ν) ∈ T(x,y)B satisfies |(x, η)|U ≤ |(y, ν)|V. We should note that the intersection of a horizontal-like disk with a vertical-like disk in the same bi-disk is not necessarily a singleton. However, if the absolute values of the slopes of each are never unity, then their intersection must be a singleton. Hubbard and Oberste-Vorth go on to show the following theorems: Theorem 6.4 (Hubbard-Oberste-Vorth). pr1 ◦ f |W1∩(U1×y) and pr2 ◦ f −1|W2∩({x}×V2) must have the same topological degree which is called the degree of the crossed mapping. Theorem 6.5 (Hubbard-Oberste-Vorth). Let . . . , B−1 = U−1 × V−1, B0 = U0 × V0, B1 = U1 × V1, . . . be a bi-infinite sequence of bi-disks, and fi : Bi →− Bi+1 be crossed mappings of degree 1 × with Ui of uniformly bounded size in Ui and Vi of uniformly bounded size in Vi. Then for all m ∈ Z, 1. The set WmS = {(xm, ym)|∃(xn, yn) ∈ Bn for all n > m such that fn(xn, yn) = (xn+1, yn+1)} is a closed vertical-like Riemann surface in Bm, and pr2|WmS : WmS → Vm is an isomorphism. 2. The set WmU = {(xm, ym)|∃(xn, yn) ∈ Bn for all n < m such that fn(xn, yn) = (xn+1, yn+1)} is a closed horizontal-like Riemann surface in Bm, and pr1|WmU : WmU → Um is an isomorphism. 53 3. Moreover, the sequence (xm, ym) := WmS ∩ WmU for m ∈ Z is the unique bi-infinite sequence with (xm, ym) ∈ Bm for all m ∈ Z, and fm(xm, ym) = (xm+1, ym+1). Additionally, the map fn|WnS : WnS → WnS+1 is a strong contraction of Poincare´ metrics and f |n f −1(WnU+1) : f −1(WnU+1) → WnU+1 is a strong expansion of Poincare´ metrics. The strong contraction and expansion is due to the fact that Ui and Vi are relatively compact in U and V, respectively. Moreover, the expansions and contractions are uniform for all n because of the uniform sizes of the Ui ’s and Vi ’s. [HOV94b] guarantees that the tangent spaces of stable and unstable manifolds are respectively in the vertical-like and horizontal-like cone fields. We will need a mild extension of this result, which was already known to Hubbard and Oberste-Vorth: Theorem 6.6. Given the same setup as Theorem 6.5, the slopes of the stable and unstable manifolds are uniformly bounded in absolute value away from 1. Moreover, this bound depends only on the sizes of the Vn ’s in the Vn’s and the sizes of the Un ’s in the Un’s, and this dependency is continuous. Proof. Any unstable manifold WnU is the forward image of WnU−1 ∩ Wn−1 by fn−1. Therefore it is completely contained in Un × Vn . Pick any point (x, y) ∈ WnU with (ζ, η) ∈ T(x,y)(WnU). WnU is the graph of a conformal map from Un to Vn , and so by Schwarz’s lemma , |(ζ, x)|Un ≥ |(η, y)|Vn . Because the Vn ’s are of uniformly bounded size in Vn, then there exists some κ > 1 so that the inclusion map ι : Vn → Vn contracts the Poincare´ metric by at least a factor of κ. Hence |(ζ, x)|Un ≥ κ · |(η, y)|Vn. The uniform bound guarantees that κ does not depend on n, but note that κ depends continuously on the maximum of the sizes of the Vn ’s in the Vn’s. 54 The proof for stable manifolds is analogous. 6.2 Horizontal Disk Contraction We need to show that horizontal disks get closer to each other under iteration of crossed mappings, but in order to do this we will need a definition of distance for horizontal disks that is adapted to this particular situation. Definition 6.7. In a bi-disk, the distance between two horizontal-like disks D1 and D2 whose slopes never have absolute value 1 relative to a vertical-like disk V is defined as: dV (D1, D2) = d pr2(D1 ∩ V), pr2(D2 ∩ V) where d in the previous equation refers to Poincare´ distance in the disk. Definition 6.8. The distance between two horizontal-like disks D1 and D2 whose slopes never have absolute value 1 is defined as: d(D1, D2) = sup dV (D1, D2) V is a verticallike disk This definition of a metric on horizontal disks satisfies the triangle inequality and the equivalence d(D1, D2) = 0 ⇐⇒ D1 = D2. Theorem 6.9. The distances between horizontal-like disks are uniformly contracted by a 1-crossed mapping. Proof. Let f : B1 →− B2 be a 1-crossed mapping. Let D1 and D1 be any two hor× izontal like disks in B1. Then their images are, respectively, D2 and D2 , which are two horizontal-like disks in B2. Choose any ε > 0. Let Z2 be any vertical-like disk in 55 B2 with dZ2(D2 , D2 ) − d(D2 , D2 ) < ε. Let Z1 = f −1(Z2), which must be a verticallike disk in B1. Because of the strong expansion for f −1 on vertical-like disks, then dZ1(D1 , D1 ) ≥ κdZ2(D2 , D2 ) for some κ > 1. So, d(D1 , D1 ) ≥ dZ1(D1 , D1 ) ≥ κdZ2(D2 , D2 ) ≥ κ d(D2 , D2 ) − ε Since this is true for every ε > 0, then d(D2 , D2 ) ≤ 1 κ d(D1 , D1 ). Theorem 6.10. The metric d on the space of horizontal-like disks is equivalent to the metric d defined as: d (D1, D2) = sup d{x}×D(D1, D2) x∈D Proof. d (D1, D2) ≤ d(D1, D2) is clear because d considers only a subset of the verticallike disks that d does. Choose any x1 ∈ D. Suppose a vertical-like disk Z intersects D1 at (x1, y1) and D2 at (x2, y2). Because Z is vertical-like, then d(x1, x2) ≤ d(y1, y2). There is another point (x1, y3) on D2 and because D2 is horizontal-like, then d(y2, y3) ≤ d(x1, x2). The triangle inequality gives that d(y1, y3) ≤ d(y1, y2) + d(y2, y3) ≤ d(y1, y2) + d(x1, x2) ≤ 2d(y1, y2) = 2 · dZ(D1, D2). If we take the supremum over all Z that intersect D1 at a point with x-coordinate x1 and then take the supremum over all x1 ∈ D, then we get that d (D1, D2) ≤ 2 · d(D1, D2). Theorem 6.10 tells us that convergence of horizontal disks under our metric is equivalent to uniform convergence in vertical slices. 56 6.3 Perturbations of One-Dimensional Orbit Portraits Fix any one-dimensional quadratic wake W, and choose any c ∈ int (W). We have an actual orbit portrait O of fc associated with W. We have the non-fattened puzzle pieces {Πi} with their associated Markov graph Γ. We also have the fattened puzzle pieces {∆i} associated with O, along with the relations that ∆ j is relatively compact in fc(∆i) when we have containment of the corresponding non-fattened puzzle pieces: Π j ⊂ fc(Πi). If fc is a small enough perturbation of fc in the C1-topology, then it is clear that ∆ j is relatively compact in fc (∆i). We will define puzzle pieces in two dimensions that will code some, but not all of the points in the Julia set. This coding will be valid throughout an open region of parameter space, which we will describe. The one-dimensional fattened puzzle pieces are bounded. Let D(0, Rc) be an open disk centered at zero that is large enough to contain the closures of all of the fattened puzzle pieces. Definition 6.11. For every fattened puzzle piece ∆i, define Bi = ∆i × D(0, Rc). The Bi’s will be called the two-dimensional (fattened) puzzle pieces. Theorem 6.12. There exists some positive constant εWc , depending only on c and W (and continuously on c), such that whenever 0 < |b| < εWc and Πi → Π j is in Γ (and Πi is non-critical), then Hb,c : Bi →− Bj is a one-crossed mapping. × Proof. Suppose Πi → Π j is in Γ. Because there are finitely many arrows in Γ, it suffices to prove the statement for a single pair of fattened puzzle pieces. The degenerate He´non mapping H0,c maps all of C2 to the co-dimension 1 complex parabola x = y2 + c and reduces to the one-dimensional dynamical system x → x2 + c in 57 the first co-ordinate. Thus pr1(Bj) = ∆ j is relatively compact in pr1 ◦ H0,c(Bi) = fc(∆i). H0−,1c(Bj) is the union of two infinitely tall open cylinders. They each have the form fc−1(∆ j) × C, where one cylinder has one branch of the inverse image fc−1 and the other cylinder has the other branch. Exactly one of these cylinders intersects Bi, and in fact, the projection to the first coordinate of this cylinder is the appropriate branch of fc−1(∆ j) and is relatively compact in ∆i = pr1(Bi). Also the disk D(0, Rc) was chosen precisely so that pr2 ◦ H0,c(Bi) = ∆i is relatively compact in pr2(Bj) = D(0, Rc). pr2 ◦ Hb,c is uniformly continuous in the C1 topology in b and c inside a bounded region of dynamical space. Note that pr1 ◦ Hb−,1c(x, y) = y, and does not depend on the parameters at all. We have shown that these relations of relative compactness are preserved under small perturbations of parameters. Thus, there is some small open domain around (0, c) so that (b , c ) in this domain implies that pr1 Hb−1,c (Bj) ∩ Bi is relatively compact in pr1(Bi), pr2 Hb ,c (Bi) ∩ Bj is relatively compact in pr2(Bj), Hb ,c maps the vertical boundary of Bi outside the closure of Bj, and Hb−1,c maps the horizontal boundary of Bj outside the closure of Bi. For any (b , c ) in this region, we will show explicitly how Hb ,c : Bi →− Bj is realized × as a crossed mapping. Let W1 = Bi ∩ Hb−1,c (Bj). Let W2 = Hb ,c (W1) = Hb ,c (Bi) ∩ B j. Let U1 = π1(W1) = π1 Bi ∩ Hb−1,c (B j) . Let V2 = π2(W2) = π2(Hb ,c (Bi) ∩ B j). We’ve already shown that U1 is relatively compact in ∆i and V2 is relatively compact in D(0, Rc). Hb ,c : W1 → W2 is a holomorphic isomorphism. Pick any y ∈ D(0, Rc). Take any compact set K1 ⊂ ∆ j. Then K1 is closed and there is some annulus A1 that separates K1 from the boundary of ∆ j. Because Hb ,c 58 maps the vertical boundary of B1 outside B2, then pr1 Hb ,c (W1 ∩ (∆i × {y})) = ∆ j. Call the projection from Hb ,c (W1 ∩ (∆i × {y})) to its first coordinate ρ1. ρ−11 maps K1 and A1 to the disk Hb ,c (W1 ∩ (∆i × {y})), with ρ−11(A1) separating ρ−11(K1) from the boundary. Then Hb−1,c (ρ−11(A1)) separates Hb−1,c (ρ−11(K1)) from the boundary of W1 ∩ (∆i × {y}). Hb−1,c (ρ−11(K1)) is closed because it is the inverse image of a closed set under a continuous map. Hence Hb−1,c (ρ−11(K1)) is compact in W1 ∩ (∆i × {y}). This implies that pr1 ◦ Hb ,c |W1∩(∆i×{y}) : W1 ∩ (∆i × {y}) → ∆ j is proper. This map is, in fact, an isomorphism. Pick any x ∈ ∆ j. Take any compact set K2 ⊂ D(0, Rc). Then K2 is closed and there is some annulus A2 that separates K2 from the boundary of D(0, Rc). Because Hb−1,c maps the horizontal boundary of B2 outside B1, then pr2 Hb−1,c (W2 ∩ ({x} × D(0, Rc))) = D(0, Rc). Call the projection from Hb−1,c (W2 ∩ ({x} × D(0, Rc))) to its second coordinate ρ2. ρ−21 maps K2 and A2 to the disk Hb−1,c (W2 ∩ ({x} × D(0, Rc))), with ρ−21(A2) separating ρ−21(K2) from the boundary. Then Hb ,c (ρ−21(A2)) separates Hb ,c (ρ−21(K2)) from the boundary of W2 ∩ ({x} × D(0, Rc)). Hb ,c (ρ−21(K)) is closed because it is the inverse image of a closed set under a continuous map (Hb−1,c is continuous). Hence Hb ,c (ρ−21(K)) is compact in W2 ∩ ({x} × D(0, Rc)). This implies that pr2 ◦ Hb−1,c |W2∩({x}×D(0,Rc)) : W2 ∩ ({x} × D(0, Rc)) → D(0, Rc) is proper. This map is, in fact, an isomorphism. Hence (W1, W2, Hb ,c ) is a degree one crossed mapping from B1 = ∆i × D(0, Rc) to B2 = ∆ j × D(0, Rc). We have shown that Hb,c : Bi →− Bj is a one-crossed mapping × in a small neighborhood around each (0, c) with c ∈ int (W), and the statement of the theorem follows. Definition 6.13. Let RW = (b, c) c ∈ W and 0 < |b| < εWc . Lemma 6.14. Given any bi-itinerary I adapted to Γ and any (b, c) ∈ RW, there is exactly one point which has this itinerary under Hb,c relative to the two-dimensional 59 puzzle pieces. Proof. There is a one-crossed mapping between any two consecutive pairs of puzzle pieces in an itinerary adapted to Γ, so a bi-itinerary adapted to Γ gives a bi-infinite sequence of 1-crossed mappings. By [HOV94b], there is exactly one point which satisfies this bi-itinerary. Definition 6.15. Let I be a bi-itinerary adapted to Γ. Then for (b, c) ∈ RW, define ΨWb,c(I) to be the unique point with this bi-itinerary for the map Hb,c. Definition 6.16. Define XWb,c = ΨWb,c I I is a bi-itinerary adapted to Γ . Theorem 6.17. For (b, c) ∈ RbW,c, then XWb,c ⊂ Jb,c. Proof. It is clear that XWb,c ⊂ Kb,c. We will show that arbitrarily close to any point of XWb,c, there are points which escape to infinity in forwards and backwards time. Fix (b, c) ∈ RbW,c. Define S+ = (x, y) |x| ≥ Rb,c and |y| ≤ |x|2 − |c| |b| − |x| S − = (x, y) |y| ≥ Rb,c and |x| ≤ |y| Some algebraic manipulation gives us that if (x, y) ∈ Hb,c(S +), then |x| ≥ |y| ≥ Rb,c and hence Hb,c(S +) ⊂ Vb+,c ⊂ Ub+,c. Thus S + ⊂ Ub+,c. It is also clear that S − = Vb−,c ⊂ Ub−,c. Choose y0 ∈ R+. Let 1+ r+ = r− = y0 1 + 4(|c| + |b| y0) 2 60 For large enough y0, then S + ∩ (C × {y0}) is a plane with a disk of radius r+ cut out of it and S − ∩ (C × {y0}) is a disk of radius r−. Also for large enough y0, then r− > r+, and in fact the ratio r−/r+ goes to infinity as y0 increases. For large enough y0, we see that S + ∩ S − ∩ (C × {y0}) is an annulus entirely in Ub,c with a modulus as large as we please. fc is conjugate near infinity to x → x2 with a conjugating Bo¨ttcher map that is tangent to the identity at infinity. Let Cg be the level curve where the Green’s function is equal to g. Because of the tangency of the Bo¨ttcher map at infinity, then Cg approximates a circle as g → ∞ and there must exist some y0 > Rb,c and g so that Cg ⊂ int (π1(S + ∩ S − ∩ (C × {y0}))), where π1 is projection to the first coordinate. The choice to make the one-dimensional fattened puzzle pieces of fc have an outer boundary where the Green’s function is 1 was arbitrary. We can choose for the outer boundary the level curve Cg. The choice for the vertical component of the twodimensional fattened puzzle pieces to have radius Rc was also arbitrary. We could have chosen any disk that contains all of the one-dimensional fattened puzzle pieces. Let us choose for the vertical component of the two-dimensional fattened puzzle pieces the disk D(0, y0). Notice that we ensured that Cg is relatively compact in D(0, y0). The entire vertical portion of the boundary of each two-dimensional puzzle piece is contained in the closed set S +. Because for any y ∈ D(0, y0), Cg ⊂ int (π1(S + ∩ (C × {y}))) and Cg is compact, then there exists a finite distance Th so that the Poincare´ distance in any unstable manifold in any two-dimensional puzzle piece between any point of XWb,c and S + is less than Th.1 1This depends on the fact that the set all points of XWb,c whose itinerary at the 0th index is the fattened puzzle piece Bi is relatively compact in Bi. This property is due to the fact that whenever Πi → Π j is in Γ, then we have a 1-crossed mapping Hb,c : Bi → Bj, and the corresponding V2 (see Definitition 6.1) for this crossed mapping is relatively compact in the corresponding V2. Also there are finitely many arrows. We also depends on the fact that the space of vertically-bounded horizontal-like disks is compact, and we know that the unstable manifolds occupy a relatively compact subset of the disk when projected to their second coordinate (see the proof of Theorem 6.6 for details). 61 Choose any ε > 0. There is a uniform expansion λh > 1 on the unstable manifolds. Let n be an integer larger than logλh(Th/ε). To realize a point of Ub+,c which is ε-close to any x ∈ XWb,c, we need only to look in the two-dimensional puzzle piece Bi which is at index n in the itinerary of x. Bi contains a point u ∈ Ub+,c which is in the unstable manifold of Hb◦,nc(x) and is at distance at most Th from Hb◦,nc(x). Hence x is at most a distance of ε from Hb◦,−cn(u) ∈ Ub+,c. The entire horizontal portion of the boundary of every two-dimensional puzzle piece is entirely contained within S −, and in fact, there is some finite distance Tv so that for every x in the closure of any of the one-dimensional fattened puzzle pieces, then the Poincare´ distance in any stable manifold in any two-dimensional puzzle piece between any point of XWb,c and S − is less than Tv. As we did for Ub+,c, using the uniform contraction on stable manifolds, one can show that points of Ub−,c are arbitrarily close to every point of XWb,c. Hence we see that XWb,c ⊂ Kb,c ∩ Ub−,c ∩ Ub+,c. 6.4 Continuity of XWb,c Lemma 6.18. If I1 and I2 are distinct bi-itineraries adapted to Γ, then ΨWb,c(I1) ΨWb,c (I2 ). Proof. We will give a proof by contradiction. Let x0 = ΨWb,c(I1) = ΨWb,c(I2) with I1 I2. Let xi = Hb◦,ic(x0). I1 and I2 differ at some index n. Thus xn is in the intersection of two distinct two-dimensional fattened puzzle pieces which must be of the form D j × D(0, Rc), where D j is one of the small one-dimensional disks around a point of the 62 repelling cycle of fc associated with the wake W and D(0, Rc) is a disk as defined earlier. So I1 and I2 must differ for every index m > n. For indexes after n, one of these must consist of only open puzzle pieces, and these puzzle pieces must be adjacent to the successive points in the periodic orbit. Hence, one of I1 and I2 has a degenerate tail and is not adapted to Γ. Corollary 6.19. XWb,c is isomorphic to the space of bi-itineraries adapted to Γ. Proof. Since XWb,c is the image under ΨWb,c of all bi-itineraries adapted to Γ, and we know this map is injective. Theorem 6.20. XWb,c is continuous in b and c for (b, c) ∈ RW. Proof. Every point of XWb,c corresponds to a bi-itinerary I of puzzle pieces adapted to Γ. We will show that ΨWb,c(I) is continuous in b and c. Choose some point (x, y) ∈ XWb,c with an itinerary of two-dimensional puzzle pieces (. . . , Bi−1, Bi0, Bi1, . . .) adapted to Γ (where every Bin is a two-dimensional fattened puzzle piece for the map Hb,c). Let V be an open subset of RW that contains (b, c) such that all of the 1-crossed mappings between the puzzle pieces for the parameters (b, c) are still 1-crossed mappings under Hb ,c . Let V be a relatively compact open subset of V which contains (b, c) and is also bounded. It is clear that such an open sets exist. Choose (b , c ) ∈ V . We have a bi-infinite sequence of 1-crossed mappings for Hb ,c . Thus, there is vertical-like topological disk WbS ,c ⊂ Bi0 of points which have a forward itinerary (Bi0, Bi1, . . .) under the dynamics of Hb ,c . There is also a horizontal-like topological disk WbU,c ⊂ Bi0 of points which have a backwards itinerary (. . . , Bi−1, Bi0) under Hb ,c . 63 There is a unique point (x , y ) ∈ WbS ,c ∩ WbU,c with the same bi-itinerary under Hb ,c as (x, y) has under Hb,c. We will show that these disks and their intersection cannot be far away from (x, y). Let WbU,c, j be the unstable manifold in Bj for the map Hb,c. Similarly let WbU,c , j be the unstable manifold in Bj for the map Hb ,c . Define D j,k = Hb◦k,c (WbU,c,− j), which is the forward image under Hb ,c of an unstable manifold of Hb,c. This is analogous to mapping backwards by Hb−,1c and then forwards by Hb ,c as if we were attempting to conjugate the two maps by invoking the standard limit construction of the conjugating map. Every D j,k is a horizontal-like disk in Bik−j. Also WbU,c ,0 = lim j→∞ D j, j in the metric we defined for horizontal disks. Because Hb,c is a Lipshitz function of b and c on compact subsets of parameter space and because the Euclidean metric is equivalent to the Poincare´ metric on Bi−j on compact subsets of the latter, then the distances between D j,0 and D j+1,1 are uniformly bounded by some q0 · (b, c) − (b , c ) for all (b , c ) ∈ V . Mapping forward contracts distances between horizontal-like disks (uniformly so on V ), so there is some q1 < 1 so that d(D j,k+1, D j+1,k+2) < q1 · d(D j,k, D j+1,k+1). Hence d(D j, j, D j+1, j+1) < q1j q0 · (b, c) − (b , c ) and so d(WbU,c,0, WbU,c ,0) < q0/(1 − q1) · (b, c) − (b , c ) An analogous argument for stable manifolds gives: d(WbS,c,0, WbS ,c ,0) < q0 /(1 − q1 ) · (b, c) − (b , c ) The two disks WbU,c ,0 and WbS ,c ,0 have a single intersection point (x , y ) which is the unique point with bi-itinerary (. . . , Bi−1, Bi0, Bi1, . . .) under the map Hb ,c . So Ψb ,c (I) = (x , y ). Let dBi0 be the Kobayashi metric in Bi0. The geometry of the situation forces: √√ dBi0 (x, y), (x , y ) ≤ 2 · d(WbU,c,0, WbU,c ,0) + 2 · d(WbS,c,0, WbS ,c ,0) 64 Let q0 = max {q0, q0 } and q1 = max {q1, q1 }. q0 and q1 do not depend on (b , c ). We get for all (b , c ) ∈ V the inequality: √ dBi0 (x, y), (x , y ) 2 ≤ 2 · q0 1 − q1 (b, c) − (b , c ) Because the Kobayashi metric is equivalent to the Euclidean metric on compact subsets, this shows that Ψb,c(I) is locally Lipshitz and hence a continuous function of (b, c) ∈ RW. 6.5 Coding XWb,c Theorem 6.21. lim b→0 XWb,c = (x, y) y ∈ fc XWc and fc(y) = x In particular: pr1 lim b→0 XWb,c = pr2 lim b→0 XWb,c = fc XWc And points with bi-itineraries adapted to Γ (. . . , Bi−1, Bi0, Bi1, . . .) in XbW,c will limit as b → 0 and then project to a point of XWc with itinerary (Πi0, Πi1, . . .). Proof. Take any bi-itinerary I = (. . . , Bi−1, Bi0, Bi1, . . .). We will show that ΨWb,c(I) has a continuous extension to {0} × int (W), which is essentially a one-dimensional wake living inside of parameter space for He´non mappings. Choose c ∈ int (W). Let M be larger than the radius of the Julia sets in some neighborhood of (0, c). The first coordinate of orbits of XWb,c by Hb,c in this neighborhood are (M · b)-pseudo-orbits of fc. Hence, because the one-dimensional fattened puzzle pieces are open (and fc’s derivative is bounded on compact subsets), then the smaller b is, the longer that ΨWb,c(I) and pr1(ΨWb,c(I)) will have corresponding forward itineraries under 65 the actions of Hb,c and fc, respectively. Hence pr1 limb→0 ΨWb,c(I) will have an itinerary of (Πi0, Πi1, . . .). Therefore, it must be the unique point of XWc with this itinerary. The rest of the statement of the theorem follows easily. Theorem 6.22. XWb,c, Hb,c l←i−m− XWc , fc Proof. XWc , fc is isomorphic to one-sided itineraries adapted to Γ. XWb,c, Hb,c is isomorphic to bi-itineraries adapted to Γ . The union of the non-critical one-dimensional puzzle pieces has two connected components. Non-critical one dimensional puzzle pieces can be coded by which side of the critical puzzle piece they lie on. The two-dimensional puzzle pieces inherit this coding, so points in XWb,c have two-symbol codings coming from their two-dimensional puzzle piece itineraries. In one dimension, Julia sets taken from the exterior of the Mandelbrot set have a two-symbol coding derived from cutting up the plane along the inverse image of the dynamical ray that hits the critical value. If, additionally, we remove from our consideration R+, then we can unambiguously label the side of this cutting which contains the β-fixed point with an A and the side with the α-fixed point with a B. Points in Jb,c for (b, c) ∈ HOV also have a two-symbol coding arising from their realization as the inverse limit system of a one-dimensional Julia set taken from the exterior of the Mandelbrot set. If we remove the hypersuface C×R+, then on the negative real axis we can unambiguously label the region where x is positive with A and where x is negative with B and extend this coding unambiguously to HOV\R+ ×C continuously. Inside of HOV ∩ RW, these two codings agree on XWc . To see this, note that the dynamical ray that hits the critical value must lie inside the critical value puzzle piece. 66 Therefore, its inverse image must lie inside the critical puzzle piece, so for points in XWc , the two codings are identical, because XWc has no points in the critical puzzle piece. Theorem 6.23. A point of XWb,c is determined by its two-sided W-itinerary. Proof. A bi-infinite itinerary on two symbols gives a point in the inverse limit system of XWc , which is isomorphic to XWb,c. We see here that the itinerary relative to puzzle pieces has much redundant information. It is possible to identify points using a far more coarse partition of space. While it is true that there is not a well-defined two-symbol encoding on the whole Julia set throughout RW, we can give a two-symbol coding to the points of XWb,c throughout RW. Theorems 6.22 and 5.6 together guarantee that the same set of two-symbol codings is realized by XWb,c everywhere throughout RW. Hence everywhere inside RW, XWb,c has exactly one point with a particular bi-infinite AB-coding if and only if that coding has no substring equal to the characteristic kneading sequence of a wake conspicuous to W. The reason we need multiple codings is that W-itineraries are valid in HOV, but not in RW, but paths adapted to Γ give a coding in RW, but not in HOV. Definition 6.24. Let ΣW be the set of two-symbol W-bi-itineraries realized by points of XWb,c . Theorem 6.25. ΣW is a subshift of finite type where the forbidden strings are the characteristic kneading sequences for the wakes conspicuous to W. Proof. Theorem 6.22 and Corollary 5.24. 67 6.6 Relationships Between Points and Itineraries The following commutative diagram illustrates relationships between points of XWb,c, Witineraries, and itineraries and multi-itineraries relative to the fattened puzzle pieces. {} {{{{{{f{{{{{{{{{{=ΣW bEEEEEEEEgEEEEEEEEE XWb,c o Bi-itineraries of fat 2-D puzzle c pieces adaO pted to Γ d  Multi-bi-itinerary of fat 2-D puzzle pieces adapted to Γ We describe these maps: e b  / Bi-itinerary of puzzle pieces adapted to Γ b) The correspondence between puzzle pieces and their 2-D counterparts gives a trivial correspondence between bi-infinite sequences adapted to Γ. c) An itinerary of 2-D puzzle pieces gives a unique point of XWb,c using crossed mappings. d) Points of XWb,c have multi-itineraries with respect to the 2-D puzzle pieces. e) Theorem 5.9. f) XWb,c → ΣW is given by itineraries relative to the unions of the two-dimensional puzzle pieces on each side of the critical puzzle piece. ΣW → XWb,c is given by Theorem 6.23. g) Two-dimensional puzzle pieces are on one side or the other of the critical puzzle piece. 68 CHAPTER 7 MONODROMY INVARIANT Definition 7.1. Let UW = HOV ∪ RW \ (C × R+). We are defining UW to be the region where we either know we have a horseshoe by [HOV94a] or we know that we have the structurally stable sparse set XWb,c, but we are disallowing the class of loops homotopic to γc by removing the hypersurface C × R+. The reason for disallowing this is that γc has the nontrivial action on the Julia set in HOV of permuting the two symbols. We wish to isolate a region of parameter space where there is a trivial monodromy action on a particular subset of the Julia set. Points of the Julia set can be followed continuously through HOV. XWb,c is a subset of the Julia set for (b, c) ∈ RbW,c ∩ HOV. These points can be followed smoothly throughout UW. Definition 7.2. Define YbW,c = XWb,c for (b, c) ∈ RW and continuously extend YbW,c ⊂ Jb,c for (b, c) ∈ UW. We can extend YbW,c because it is part of the Julia set, and the Julia set can be followed continuously inside of HOV. Also this extension is well-defined because YbW,c has a trivial monodromy around γb, the loop that generates the fundamental group of UW. Lemma 7.3. If we fix a wake W, the W-itineraries realized by YbW,c are constant for (b, c) ∈ UW. Proof. They are constant in RW because the W-itineraries of points in XWb,c are constant in this region. In HOV, points of J are identified with their itinerary and move continuously with respect to the parameters, and the A and B regions do not interchange in UW. 69 Choose any class of loops γ ∈ Π1(UW ∩ HC, p0), where p0 = (b0, c0) ∈ HR is a parameter value for a real horseshoe map. Lemma 7.4. γ has a trivial monodromy action on YW b0,c0 . Proof. YbW,c is structurally stable in UW. The fundamental group of UW is generated by γb, which has a trivial monodromy action. The monodromy action ρ(γ) gives a continuous automorphism ρ(γ) of the full 2shift. Theorem 7.5. If α ∈ Σ2 does not contain as a substring the characteristic kneading sequence of any of the finitely many wakes conspicuous to W, then ρ(γ) acts trivially on α. Proof. For any (b, c) ∈ UW, the set YbW,c is precisely the set of points of Jb,c whose itinerary does not include the characteristic kneading sequence of any wake conspicuous to W as a substring. Since γ has a trivial monodromy action on YW b0,c0 , then ρ(γ) must act trivially on any point which does not contain as a substring the characteristic kneading sequence of any wake conspicuous to W. A somewhat stronger statement than theorem 7.5 is actually true. The points in YbW,c exist and are a subset of J throughout UW whether or not the parameter values are inside the horseshoe locus. We can thus follow some points of J as we pass through non-hyperbolic parameter values. 70 CHAPTER 8 MONODROMY CONJECTURES In order to describe our conjectures for the monodromy action, we will need some background and vocabulary arising from Koch’s computer-assisted investigation of He´non parameter space ([Koc05] and [Koc07]). 8.1 Speculative Structure of He´non Parameter Space When the Jacobian of a quadratic He´non mapping is zero, the first co-ordinate of the mapping reduces to the one-dimensional quadratic mapping. Koch discovered using Karl Papadantonakis’ program SaddleDrop that when taking c-plane slices of complex He´non parameter space, as one moves away from the degenerate b = 0 case, then different renormalized Mandelbrot sets strictly contained in the Mandelbrot set break off and move in different directions. Moreover, she found that the direction they move in is tied to the kneading sequence of the polynomials whence they originated in the Mandelbrot set. Specifically, the direction depends most on the digits immediately preceding , or in other words at the end of the finite representation of kneading sequences. Renormalized Mandelbrot sets with a kneading sequence ending in an A generally move in the direction in the c-plane that the parameter value b is perturbed in, and those with a kneading sequence ending in B generally move in the opposite direction as b does. As one perturbs b away from 0, then in c-plane slices, the Mandelbrot set seems to split into two different “herds”. One contains all renormalized Mandelbrot sets that have a kneading sequence ending in A and the other those that end in B. As one perturbs b even farther away from zero, then each of these herds split up based on the digit in the kneading sequences second from the end. Then these split by the third to last 71 digit in the kneading sequences. This dyadic splitting phenomenon has been witnessed experimentally using SaddleDrop to a depth of 5 splits. Misiurewicz points split into a Cantor set worth of points and appear in every herd. This is in direct contrast to the hyperbolic components, which follow only one herd. This is because in parameter space, the boundary of the region where there is an attracting periodic point is the solution of an algebraic curve and thus can only interect a plane in finitely many points. As noted earlier, the herd of Mandelbrot sets with a kneading sequence ending in A moves in the general direction that b does, and the herd of Mandelbrot sets with a kneading sequences ending in B moves in essentially the opposite direction. The herd of Mandelbrot sets with kneading sequence ending in AA moves a bit farther in the direction of b than does the herd with sequences ending in BA. Additionally, the two herds associated with AB and BB also move differentially, but the presence of the first B in the kneading sequence flips the direction that the hierarchically lower-level herds move in, so the BB herd is perturbed slightly more in the direction of a than is the AB herd. If one perturbs b slightly in the positive real direction, reading from left to right, one would expect to see the eight herds obtained after three splittings associated with kneading sequences in the following order: AAB, BAB, BBB, ABB, ABA, BBA, BAA, AAA. 8.2 Monodromy Conjecture In order to describe our conjecture for the monodromy action, we will need to develop a language for conveniently describing a class of continuous automorphisms of the full 2-shift that commutes with the shift operator. Definition 8.1. A compound marker endomorphism is a mapping on Σ2 described by a 72 finite collection of finite strings on two symbols as well as a . Each string has a single . We use this finite collection of strings to define a mapping on Σ2 by the following algorithm: If the sequence matches any of the strings at any location (where the can match anything), then the image of this sequence will have the opposite letter in the position that matched . If a letter at a position does not match any string at the position, it is left unchanged. A compound marker endomorphism is always a continuous endomorphism of Σ2 that commutes with the shift. It is not always bijective, though. Definition 8.2. If a compound marker endomorphism is an automorphism, we call it a compound marker automorphism. Compound marker automorphisms are a generalization of marker automorphisms. Interestingly enough, there are compound marker automorphisms whose component strings are not individually compound marker automorphisms themselves. The author knows of no automorphism of the full 2-shift that is not a composition of compound marker automorphisms and the shift. It would be of great interest to either prove or disprove that these generate the automorphisms of the full two-shift. We conjecture the following on the basis of computer experimentation: Conjecture 8.3. Suppose that γ ∈ Π1 H0C, (b0, c0) is such that γ winds around a herd corresponding to a given string x, and this herd comes from a wake W1 with conspicuous sub-wakes W2, . . . , Wn. Let yi = K(Wi). Then ρ(γ) is the following compound marker endomorphism: x y1, . . . , x yn Note that in this conjecture, another way to view the yi’s is as the initial segments of 73 the kneading sequences of polynomials in the region of the Mandelbrot set whence the herds originated. If Conjecture 8.3 were true, then no point of J could be permuted by a loop in UW unless that point had in its itinerary the characteristic kneading sequence of W or that of some wake contained in W. That is what we proved in Chapter 7. As the herds split and move away from each other, hyperbolic components from the Mandelbrot set will follow one herd, and create a gap in the herd that they do not follow. These gaps give rise to loops in the Horseshoe locus. It is unknown if all loops in the horseshoe locus are generated by going through these gaps (along with the two generators from HOV). We have another conjecture, which we also make on the basis of computer experimentation: Conjecture 8.4. If the compound marker endomorphism predicted by Conjecture 8.3 is not an automorphism, then no loop could wind around the herds in question and only those herds while staying inside the horseshoe locus. We are conjecturing a connection between the algebraic structure of automorphisms of the full 2-shift and the topological structure of the horseshoe locus in parameter space of He´non mappings. 74 CHAPTER 9 EXAMPLES We will conclude with a few examples which demonstrate various aspects of Theorem 7.5, Conjecture 8.3, and Conjecture 8.4. All of the following pictures of parameter space were created using the computer program SaddleDrop, written by Karl Papadantonakis. The program is available at the website http://www.math.cornell.edu/ dynamics/ and is an accompaniment to [HP00]. Generically for quadratic He´non mappings, there are two fixed points, and the map has a larger eigenvalue (in absolute value) at one of these fixed points. Saddledrop displays a parameterization of the unstable manifold at this fixed point, and allows the user to select critical points of G+ restricted to this unstable manifold. Saddledrop then smoothly follows these points throughout parameter space wherever it is possible to continue them. Saddledrop draws planar slices of parameter space and colors points according to the value of G+ at the identified critical points. In the following pictures, we follow the 16 most prominent critical points. There is no good way to chromatically represent the 16 rates of escape in one picture, but Saddledrop gives us three ways to combine this information to color points. In the following pictures, the rate of escape of the slowest escaping critical point is represented as a color at that pixel. This “slowest escaping” way of coloring parameter space colors non-hyperbolic parameter values in darker hues. We do not catch all non-hyperbolic points with our 16 test critical points, but at this resolution, the pictures do not change significantly with the addition of more critical points. Definition 9.1. We say that a marker string matches a bi-infinite sequence if some shift of the marker string is such that in each position where the marker string has an A or a B, then the symbol in the marker string matches the symbol in the bi-infinite sequence. 75 We say that the marker string matches the sequence at the position where the symbol is. Definition 9.2. We say that a marker string matches another marker string at a given position if a shift of the first string is such that in each position where both marker strings have either an A or a B, then the two marker strings have the same symbol. We say that the first marker string matches the second marker string at the position on the second marker string where the symbol is. 9.1 B BAA In the following pictures of c-plane slices of He´non parameter space, we will perturb the Jacobian in the positive imaginary direction, and we will see that the “tail” of the Mandelbrot set splits into two parts and that the airplane follows the top part (Figures 9.1 to 9.7). The Jacobian in this sequence of pictures goes from 0 to .05i. 76 Figure 9.1: Parameter slice with b = 0 77 Figure 9.2: Parameter slice with b = 0.005i 78 Figure 9.3: Parameter slice with b = 0.01i 79 Figure 9.4: Parameter slice with b = 0.015i 80 Figure 9.5: Parameter slice with b = 0.02i 81 Figure 9.6: Parameter slice with b = 0.03i 82 Figure 9.7: Parameter slice with b = 0.05i 83 Let Wair be the one-dimensional parameter wake with boundary rays at angles 3/7 and 4/7 and which is associated with the airplane polynomial. The only wake conspicuous to Wair is itself, and K(Wair) = BAA. There is a gap in the bottom herd left by the airplane because the airplane traveled with the top herd. Every polynomial in Wair (except for the airplane component itself) has an initial kneading sequence BAA, and this region is where the herd we loop around comes from. The airplane follows the A-herd, because its kneading sequence is BA, which ends in A, so the herd we are looping around is the B herd. Conjecture 8.3 implies that the monodromy action around the loop in Figure 9.8 is B BAA. Figure 9.8: Loop around B herd of Wair with b = 0.05i The black loop in Figure 9.1 is not in UWair, but if this loop is homotopic to a loop in UWair, then by Theorem 7.5, the monodromy action of this loop would have to be trivial on every sequence which did not include the substring BAA. This is a necessary 84 condition for the monodromy action of this loop to be B BAA. 9.2 BB BAA and AB BAA The B-herd is composed of the AB- and the BB-herds. As we perturb b farther, the BB and the AB herds split up. In the Figure 9.9, we can also see the AB and the BB herds splitting further into the AAB, BAB, ABB and BBB herds. Figure 9.9: Loops around BB and AB herds of Wair with b = 0.2 + 0.3i The automorphism B BAA is the composition of AB BAA with BB BAA, and 85 these two automorphisms commute. Conjecture 8.3 implies that the monodromy action of the purple loop around the AB herd of Wair is AB BAA and that the monodromy action of the red loop around the BB herd of Wair is BB BAA. 9.3 A BAA Lemma 9.3. A BAA is not an automorphism of the full 2-shift. Proof. A BAA maps both of the sequences BAA and BAB to the sequence BAB, so this endomorphism is not injective. The author is unable to find a a loop in the horseshoe locus that travels only around the A-herd of Wair (see Figure 9.7). There is no obvious gap to go through, because the airplane travels with the A-herd. Conjecture 8.4 implies that such a loop does not exist in the Horseshoe locus. To find a loop that goes around the A-herd of Wair, the computer experimentation suggests that one must find a gap created by some other hyperbolic component which travels in the other direction. 9.4 A BAA, A BABBA There is a period 5 renormalized Mandelbrot set on the real axis of the Mandelbrot set to the right of the airplane at the landing point of the 13/31 and 18/31 parameter rays whose kneading sequence is BABB . Call the parameter wake associated with this component WBABB. The only two wakes conspicuous to WBABB are itself and Wair, and 86 Figure 9.10: Loop around A herd of WBABB K(WBABB) = BABBA. The loop in Figure 9.10 goes through the A herd corresponding to where this period 5 component left a gap. The polynomials in M between this period 5 component and the airplane all have an initial kneading sequence of BABBA. So this loop goes around the A herd of polynomials in the Mandelbrot set with initial kneading sequences of either BAA or BABBA. Lemma 9.4. The two-string compound marker endomorphism A BAA, A BABBA is a compound marker automorphism. Proof. We will say that a marker string matches at a particular position when that position is where the matches. Let φ be the compound marker endomorphism A BAA, A BABBA. It is clear that φ is continuous and commutes with the shift. We will show that φ is bijective by showing that φ is an involution. Choose any x = (xi) ∈ {A, B}Z. Let y = (yi) = φ(x) and z = (zi) = φ(y). We will show that z = x by showing that x and y differ at an index if and 87 only if y and z differ at the same place. Suppose x and y differ at index 0. Then x−1 = x2 = A and x1 = B. Also either x3 = A (case 1) or x3 = x4 = B and x5 = A (case 2). Let us consider case 1 first. In this case, neither A BAA nor A BABBA can match x at position −1, 1, or 2, so y−1 = x−1 = A, y1 = x1 = B, and y2 = x2 = A. Either string could potentially match x at position 3. Let 1a be the case where there is no match and let 1b be the case where one of the two marker strings matches x at position 3. Let us consider case 1a. Then, in addition to the information we have for case 1, we know that y3 = x3 = A. Hence the marker string A BAA matches y at index 0. Thus, z0 y0. Let us consider case 1b. We know that y3 = B. We also know that no matter which string matches x at index 3, x4 = B and x5 = A. Neither marker string can match x at index 4 or 5, so y4 = x4 = B and y5 = x5 = A. This forces the string B BABBA to match y at index 0, and we see that z0 y0. Now let us consider case 2. Neither marker string can match x at index −1, 1, 2, 4, or 5, so y1 = x1 = B, y2 = x2 = A, y4 = x4 = B, and y5 = x5 = A. However, either marker string can match x at index 3, so y3 may be either A or B. If y3 = A, then y matches the marker string A BAA at index 0. If y3 = B, then y matches A BABBA at index 0. Either way, z0 y0. We have proven that if x0 y0, then y0 z0. Now, let us assume that y0 z0. Then either y matches B BAA at index 0 (case 3) or y matches B BABBA at index zero (case 4). Let us consider case 3. Suppose that x−1 = B A = y−1. Then x must match one of 88 the marker strings at index −1. This implies that x0 = B and x1 = A y1. So x must match a marker string at index 1 as well, but x0 = B contradicts this. So we know that x−1 = A. Now suppose that x1 = A. Then because x1 y1, then one of the marker strings must match x at index 1, so x2 = B and x3 = A. Because x2 y2, then a string must match x at index 2 as well, which implies that x3 = B, a contradiction. Thus, x1 = B. Since x1 = B, then neither string can match at index 2, so y2 = x2 = A. Let 3a be the case that x3 = A and let 3b be the case that x3 = B. Let us consider case 3a. In this case, x matches the marker string B BAA at index 0, so x0 y0. Let us consider case 3b. In this case, since x3 y3, then we know that x4 = B and x5 = A. Hence x matches the string A BABBA at index 0, and x0 y0. Let us consider case 4. Suppose that x5 = B. Because x5 y5, then a marker string must match x at index 5, so x4 = A. Because x4 y4, then a marker string must match x at index 4, so x3 = A. Because x3 y3, then a marker string must match x at index 3, so x4 = B, a contradiction. Hence we know that x5 = A. Neither string can now match x at index 4, so x4 = y4 = B. Similarly, the fact that x4 = B keeps either marker string from matching x at index 2, so x2 = y2 = A. Neither string can match x at index 1 because x2 = A, so x1 = y1 = B. Because x1 = B, then neither string can match x at index −1,so x−1 = y−1 = A. If x3 = A, then x matches A BAA at index 0, and x0 y0. On the other hand, if x3 = B, then x matches A BABBA at index 0, and x0 y0. Thus, we see that in all cases, z0 y0 implies that x0 y0. So x0 = y0 if and only if y0 = z0. In either case, x0 = z0. Marker endomorphisms commute with the shift, so x = z = φ(φ(x)) for any x ∈ Σ2. φ is its own inverse and is an automorphism. As mentioned earlier A BAA by itself is not an automorphism. However, A BAA, 89 A BABBA is in fact a compound marker automorphism, and Conjecture 8.3 implies that this automorphism describes the monodromy action of the loop in Figure 9.10. 9.5 A BAA, A BABBA, B BAA If we could move the two gaps from the loop that generated A BAA, A BABBA and the loop that generated B BAA on top of each other, then we might expect given Conjecture 8.3 that a loop going through the combined gap to generate the compound marker endomorphism A BAA, A BABBA, B BAA. However, this action fails to be an automorphism (Lemma 9.6). Figures 9.11, 9.12, and 9.13 show an attempt to move these gaps on top of each other. 90 Figure 9.11: Parameter slice with b = −0.03 + 0.02i 91 Figure 9.12: Parameter slice with b = −0.03 + 0.01i 92 Figure 9.13: Parameter slice with b = −0.03 93 This is not to say that there are not loops that go through both gaps. The monodromy action on such loops would be the two-string compound marker automorphism A BAA, A BABBA composed with the simple marker automorphism B BAA, which would be an automorphism. This loop could be contained in the b = −0.03 + 0.02i slice of parameter space, as is illustrated in Figure 9.11. Complex conjugacy is a symmetry of parameter space. If it were possible to homotope such a loop to a real Jacobian, then it would therefore also be possible to homotope that same loop to the b = −0.03 − 0.02i slice of parameter space. Conjecture 8.3 would then imply that the monodromy action of this loop would simultaneously have to be A BAA, A BABBA post-composed with B BAA as well as B BAA post-composed with A BAA, A BABBA. However, the action A BAA, A BABBA does not commute with B BAA (Lemma 9.5). Therefore, if Conjecture 8.3 is correct, then it is not possible to move these two gaps on top of each other. Lemma 9.5. The two-string compound marker automorphism A BAA, A BABBA does not commute with the one-string simple marker automorphism B BAA. Proof. Let ψ be the action of the compound marker string A BAA, A BABBA, and let ϕ be the action of the compound marker string B BAA. Let x be the bi-finite string AABABBABA. Then: ψ(x) = ABBABBABA ϕ(ψ(x)) = ABBABBBBA And we also see that: ϕ(x) = AABABBBBA ψ(ϕ(x)) = AABABBBBA Hence we see that ψ ◦ ϕ ϕ ◦ ψ, because they disagree on x. 94 Lemma 9.6. The three-string compound marker endomorphism A BAA, A BABBA , B BAA is not an automorphism. Proof. This compound marker endomorphism maps both of the bi-infinite sequences BAABABBA and BABBABBB to BABBABBB, so it cannot be injective. 9.6 ABAAB BAA and BBAAB BAA Figure 9.14 shows the ABAAB and the BBAAB herds of Wair. Here the Jacobian is −0.1 + 0.9i. ABAAB BAA and BBAAB BAA are compound marker endomorphisms that are not compound marker automorphisms. When b is large enough to separate the BBAAB herd from the ABAAB herd of Wair, other nonhyperbolic components come in between them from other parts of the Mandelbrot set. Figure 9.15 shows the same region of parameter space, but following different critical points. 95 Figure 9.14: ABAAB and BBAAB herds of Wair with b = −0.1 + 0.9i Overlaying Figures 9.14 and 9.15 yields Figure 9.16. 96 hi hi Figure 9.15: Other herds near the ABAAB and BBAAB herds of Wair with b = −0.1 + 0.9i 97 Figure 9.16: Other herds obstructing loops around ABAAB and BBAAB herds of Wair with b = −0.1 + 0.9i xx The author is unable to find a loop inside the horseshoe locus that loops only around the BBAAB herd of Wair. Conjecture 8.4 claims that such a loop does not exist. 98 APPENDIX A MONODROMIES OF INVERSE LIMIT SYSTEMS A.1 Inverse Limit System Setup This chapter will show some results that relate monodromy actions of inverse limit systems with those of the base dynamical system. Let S X and S Y be locally trivial fiber bundles with totally disconnected fibers over base spaces X and Y, respectively with natural projections τX : S X → X and τY : S Y → Y, each of which takes a fibre to its basepoint. For every x ∈ X, let sx ⊂ S X be the fiber over the point x. For every y ∈ Y, let sy ⊂ S Y be the fiber over the point y. We let π : X → Y be a continuous map and π˜ : S X → S Y also be continuous, so that the following diagram commutes: S x τX / X π˜ π  Sy τY /  Y π˜ maps fibres to fibres. In particular it maps a point z ∈ S X in the fibre above x ∈ X to a point in S Y in the fibre above π(x). Also, for any y ∈ Y, we have some dynamical system gy : sy → sy, defined continuously with respect to y ∈ Y. We also assume that gy : sy → sy is surjective for all y. We insist that there is a dynamical system fx : S x → S x which is the inverse limit of gπ(x). (S x, fx) = l←i−m−(S π(x), gπ(x)). In particular S x is isomorphic to (. . . , z−1, z0, z1, . . .) ∀i ∈ Z, zi ∈ S π(x) and gπ(x)(zi) = zi+1 . Define πˆ : S x → S π(x) be the natural projection associated with the inverse limit construction that takes (. . . , z−1, z0, z1, . . .) to z0. 99 For every x ∈ X and y = π(x), We have the following commutative diagram: sx fx / sx π˜ π˜  sy gy  / sy Define f : S X → S X by f (z) = fτX(z)(z). Similarly define g : S Y → S Y by g(z) = gτY(z)(z). f and g are dynamical systems on S X and S Y, respectively, for which every fibre is invariant. So we also have: SX f / SX π˜ π˜  SY g  / SY This setup describes a situation where we have an inverse limit of a hyperbolic dynamical system. Y should be thought of as the parameter space for the hyperbolic dynamical system, and X should be thought of as the parameter space for the inverse limit system. S x and S y are the dynamical spaces for our maps with respective parameter values x and y. A.2 Coding Setup Fix any basepoint x0 ∈ X and let y0 = π(x0) ∈ Y. Let (∆1, . . . , ∆d) be a partition of S y0. The pull-back by π˜ induces a partition (Θ1, . . . , Θd) on S x0. Θi = z ∈ S x0|π(z) ∈ ∆i . We do not insist that these partitions have the Markov property. Points in S x0 and S y0 have itineraries relative to these partitions under the actions of 100 fx0 and gy0. For z ∈ S x0, let I(z) be the two-sided itinerary of z. For x ∈ S y0, let I(z) be the one-sided itinerary of z. For any z ∈ S x0, the right-infinite tail of I(z) starting at index 0 is identical to I(π(z)). A.3 Monodromy Actions In each of the spaces X and Y, the fact that we have locally trivial fiber bundles which are completely disconnected implies by the path-lifiting property that we have monodromy actions on the fibre above any basepoint, in particular x0 and y0. Let ρx : Π1(X, x0) → Aut(S x0, fx0) and ρy : Π1(Y, y0) → Aut(S y0, gy0) be these monodromy actions. Here Aut(S x0, fx0) and Aut(S y0, gy0) represent the group of continuous automorphisms on S x0 and S y0 that commute with fx0 and gy0, respectively. Every loop in X projects to a loop in Y under π, so there is a homomorphism from Π1(X, x0) to Π1(Y, y0), which we will also denote by π. (This map is a homomorphism when we consider a composition of loops such as (γ1 ◦ γ2) to mean first going around γ2 and then going around γ1, as will be the convention here. The map is an anti-homomorphism if one makes the alternate choice.) Fix some particular loop γ ∈ Π1(X, x0) for the remainder of this appendix. Iterating a point of S x0 under the dynamical system fx0 and then computing its monodromy action by γ is the same as finding the monodromy of the point and then iterating it under the dynamics. This is because if a particular path ω through S X (which projects to a loop γ in X based at x0) connects w with z, then f (ω) must connect fx(w) with fx(z). The same argument also shows that the monodromy action of a loop in Y commutes with gy0 101 S x0 ρx(γ) / S x0 fx0 fx0  S x0 ρx(γ) / S  x0 S y0 ρy(γ) / S y0 gy0 gy0  S y0 ρy(γ) /  S y0 Similarly, it does not make a difference if we compute the monodromy of a loop γ acting on z ∈ S x0 in the top space, and then project ρx(γ)(z) down by π˜ or if we first project down both z and γ and compute the monodromy action of π(γ) on π(z) ∈ S y0. This is because if some path ω in S X connects z to ρx(γ)(z), then π(ω) in S Y connects π(z) with ρy(π(γ))(π(z)). We have the following commutative diagram: S x0 ρx(γ) / S x0 π˜ π˜ S  y0 ρy(π(γ))/ S  y0 Putting these commutative diagrams together gives: S x0 BBBBBBfxB0B! ρx(γ) S x0 /S ρx(γ) x0 BBBBBBfxB0B! /S x0 π˜ π˜ π˜  S y0 BBBBBgByB0B!  S y0 π˜ ρy(π(γ)) / S ρy(π(γ))  y0 BBBBBgByB0B! /S  y0 Theorem A.1. ρy(π(γ)) = 1 if and only if ρx(γ) = 1. Proof. ( =⇒ ) Suppose that ρy(π(γ)) = 1 and ρx(γ) 1. Then there is some z ∈ S x0 with 102 ρx(γ)(z) = w z. Since S x0 is an inverse limit system of S y0, there is some n ∈ Z such that π˜ ( fxn(z)) π˜ ( fxn(w)). Since ρy(π(γ)) = 1, then ρy(π(γ))(π˜( fxn(z))) = π˜( fxn(z)), but ρy(π(γ))(π˜ ( fxn(z))) = π˜ (ρx(γ)( fxn(z))) = π˜ ( fxn(ρx(γ)(z))) = π˜ ( fxn(w)) These equations together imply that π˜( fxn(z)) = π˜( fxn(w)), a contradiction, so the assumption that ρx(γ) 1 must have been mistaken. ( ⇐= ) Suppose that ρx(γ) = 1, but that ρy(π(γ)) 1. So there is a z ∈ S y0 so that ρy0(π(γ))(z) = w z. Because the dynamical system (S y0, gy0) is surjective, then the inverse limit system of it, (S x0, fx0), must surjectively map to it by π˜. Hence, there must be a point u ∈ S x0 so that π˜(u) = z. Since ρx(γ) = 1, then ρx(γ)(u) = u, and we get: w = ρy0(π(γ))(z) = ρy0(π(γ))(π˜(u)) = π˜(ρx(γ)(u)) = π˜(u) = z This contradicts w z. Lemma A.2. Every itinerary realized by S y0 is the right-infinite tail of some itinerary realized by a point of S x0. Proof. gy0 : S y0 → S y0 is surjective, so given any point z0 ∈ S y0, we can construct a bi-infinite sequence (. . . , z−1, z0, z1, . . .) so that zi ∈ S y0 and gy0(zi) = zi+1. This bi-infinite sequence can be considered as an element of S x0 and right-infinite tail of its itinerary is the same as that of the arbitrarily chosen z0. Lemma A.3. The right-infinite tail of any itinerary of S x0 is realized by some point of S y0 Proof. Given any bi-itinerary realized by some z ∈ S x0, it is clear that its right-infinite tail is the same as the itinerary of π˜(z) ∈ S y0. 103 Corollary A.4. The itineraries of points in S x0 and S y0 have the same allowed and forbidden finite strings as well as right-infinite sequences. Proof. By Lemmas A.2 and A.3, given any finite string or right-infinite sequence, if it is realized as the itinerary of any point of either S x0 or S y0, then it must be realized by a point from the other. Theorem A.5. ρy(π(γ)) acts on itineraries by some index-wise permutation if and only if ρx(γ) acts on itineraries by the same permutation. Proof. Let δ be any permutation on the set {1, . . . , d}. ( ⇐= ) Suppose that ρx(γ) acts on itineraries by δ, that is to say that ρx(γ) sends points with itineraries . . . , Θe−1, Θe0, Θe1, . . . to points with itineraries . . . , Θδ(e−1), Θδ(e0), Θδ(e1), . . . . For every i, ρx(γ) gives a bijection from Θi to Θδ(i). π˜ (Θi) = ∆i, so ρy(π(γ))(∆i) = ρy(π(γ))(π˜ (Θi)) = π˜ (ρx(γ)(Θi)) = π˜ (Θδ(i)) = ∆δ(i) Hence a point in S y0 with itinerary (∆ f0, ∆ f1, . . .) will be sent to a point with itinerary (∆δ( f0), ∆δ( f1), . . .). ( =⇒ ) Suppose that ρy(π(γ)) acts on itineraries by δ. For every i, ρy(π(γ)) gives a bijection from ∆i to ∆δ(i). Then a point in Θi must be sent to a point in Θδ(i) by the action of ρx(γ), and we see that ρx(γ) must induce the permutation δ on the itineraries of points in S x0 . Definition A.6. A one-sided marker string on d symbols is a finite string on d symbols, 104 except for the left-most index, which will be considered as an element of the symmetric group on d symbols. 1 One-sided marker strings on d symbols act on one-sided and two-sided sequences on d symbols by implementing their permutation on a particular index of a particular sequence when the rest of the string matches the sequence to the right of that index. Definition A.7. A one-sided marker string is called a one-sided marker automorphism if the action it induces on one-sided sequences is an automorphism and it commutes with the shift operator. Theorem A.8. ρy(π(γ)) is a one-sided marker automorphism if and only if ρx(γ) is the same one-sided marker automorphism. Proof. Suppose ρy(π(γ)) is a one-sided marker automorphism of length n. Let us con- sider a new partition of S y0 indexed by {1, . . . , d}n constructed as follows: n−1 ∆ =(e0,...,en−1) g◦y0−i(∆ei ) i=0 Again, we can lift this partition to get a partition Θ .(e0,...,en−1) Whether or not a symbol in the original itinerary is permuted depends only on that symbol and the next n − 1 symbols in the itinerary, so ρy(π(γ)) acts as a permutation on the itineraries under the new partition. By Theorem A.5, ρx(π(γ)) acts as a permutation under its new partition, so under the original partition, it must induce the same one-sided marker automorphism as ρy(π(γ). The same argument works in the other direction. 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