On dilatations of surface automorphisms

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Suppose S is a compact topological surface without boundary, oriented and connected. In \S1 we go over Thurston's classification of \emph{automorphisms} of S: continuous functions f:SS with continuous inverses. The classification provides an f# homotopic to a given automorphism f, which is either: periodic (f#n is the identity for some n≥1); or f# fixes a finite union of disjoint circles γi:S1↪S; or is what Thurston calls pseudo-Anosov (pA). In the pA case, there is a finite number of points on S whose complement carries two transverse foliations by 1-submanifolds each with a holonomy invariant measure on transverse arcs. Such a structure on S can be described in terms of a quadratic differential on a Riemann surface structure on it which we go over in \S1.2. Further, there is a number λf≥1 (called the \emph{dilatation} or \emph{stretch factor}) such that the two measured foliations are stretched and shrunk respectively by λf under f#.\ When the foliations of a pA map f can be oriented consistently in neighborhoods on S, the dilatation λf is an eigenvalue of the induced action f on the homology group H1(S;Z). The numbers λf thus satisfy monic polynomials with integer coefficients. Fried \cite{Fried} showed that λf is a unit in Z[λf] and that the Galois conjugates of λf (excepts perhaps one of ±λf−1) lie in the open annulus {zC:1/λf<|z|<λf}. Numbers satisfying these properties are called \emph{biPerron}. If the Galois conjugates of an algebraic integer λ≥1 are only in the disk {|z|<λ}, it is called a \emph{Perron} number. We describe Fried's proof and some properties of dilatations in \S1.3. In \cite[Problem 2]{Fried}, Fried asked whether some power of a biPerron unit is always a surface automorphism stretch factor. This thesis is an attempt to solve this problem. \ In \S2 we go over a well-known rectangular decomposition of a Riemann surface with a quadratic differential. When the associated foliations are orientable, the quadratic differential is the square of an Abelian differential and we study this latter structure on a Riemann surface in some detail.\ A square matrix A of non-negative entries is called \emph{mixing} if some power An has only positive entries. If only the sum A+A2+... is positive, A is \emph{ergodic}. By the Perron-Frobenius theorem, an ergodic matrix has a real eigenvalue λ>0 (called the \emph{Perron root}) bigger than all its other eigenvalues in absolute value, and whose eigenspace is 1-dimensional. If the matrix has integer entries, the Perron root is a Perron number. Lind \cite{Lind84} showed the converse that all Perron numbers are eigenvalues of ergodic integer matrices. In Chapter 3 (see \cite{BRW16}), under additional hypotheses on an erdogic matrix A with entries in {0,1} we construct a closed orientable surface S of genus gdimA/2 with a self-homeomorphism f such that the dilatation λf is the Perron root of A (which is therefore biPerron).\ On the other hand, Hamenst"adt \cite{Hamen13} showed that out of dilatations smaller than R>0 on a surface of fixed genus g, the proportion of those that have only real Galois conjugates approaches 1 as R. This suggests Fried's conjecture may be false and motivated our study of the asymptotic behavior of biPerron units which we present in chapter 4. Let Bg(R) be the set of bi-Perron units no larger than R whose minimal polynomial has degree at most 2g, and say Dg(R) is the set of dilatations no larger than R of pseudo-Anosov maps with orientable invariant foliations on Sg.\ Eskin-Mirzakhani-Rafi (2016) and Hamenst"adt (2016) independently showed that the number of periodic orbits of length less than log⁡(R) for the Teichm"uller flow on the moduli space of area one Abelian differentials on Sg grows like R4g−3/log⁡(R) as R. This is an upper bound for the number of dilatations, |Dg(R)|. We showed (with H. Baik and C. Wu) that |Bg(R)| grows like Rg(g+1)/2 as R. Since dilatations Dg(R) form a subset of biPerron units Bg(R), we get that for g≥6 the proportion of dilatations Dg(R) inside biPerron units Bg(R) approaches 0 as R. Our result does not disprove Fried's conjecture.\ For the lower genera (2, 3, 4 and 5) we get an interesting result as well. If there are exactly n closed geodesics in Moduli space of the same length log⁡(λ), we say n is the \emph{multiplicity} of each of these geodesics. For geodesics bounded in length, the proportion of those closed geodesics that have multiplicity greater than some positive integer k approaches 1 as the bound in length approaches . That is, asymptotically almost all geodesics in Moduli spaces for these genera have arbitrarily high multiplicities.

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Mathematics; BiPerron; Dilatation; Pseudo-Anosov


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Hubbard, John Hamal

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Manning, Jason F.
Hatcher, Allen E

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Ph. D., Mathematics

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Doctor of Philosophy

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Attribution 4.0 International


dissertation or thesis

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