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## Topological Models For Hyperbolic And Semi-Parabolic Complex HéNon Maps

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**Author**

Radu, Remus

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**Abstract**

Consider the parameter space P[lamda] [SUBSET OF] C2 of complex H´ non maps e Hc,a ( x, y) = ( x2 + c + ay, ax), a 0 which have a fixed point with one eigenvalue a root of unity [lamda] = e2[pi]ip/q ; this is a parabola in a2 . Inside the parabola P[lamda] , we look at those H´ non maps that e are small perturbations of a quadratic polynomial p with a parabolic fixed point of multiplier [lamda]. We prove that there is an open disk of parameters (inside P[lamda] ) for which the semi-parabolic H´ non map is structurally stable on the Julia sets e J and J + . The set J + is homeomorphic to an inductive limit of J p x D under an 2 z appropriate solenoidal map [psi] : J p x D [RIGHTWARDS ARROW] J p x D, [psi]([zeta], z) = p([zeta] ), [zeta] [-] , where p ([zeta] ) J p is the Julia set of the polynomial p. The set J is homeomorphic to a solenoid with identifications, hence connected. We also consider the class of H´ non maps that are small perturbations of a e hyperbolic (or parabolic) polynomial p( x) = x2 + c. We describe the set J + as the quotient of 3-sphere with a dyadic solenoid removed by an equivalence relation. We define a lamination for the H´ non map by lifting the Thurston lamination e of the polynomial p from the closed unit disk to the unit 4-ball in C2 , using the inductive limit. "Lifting" the leaves of the lamination of the polynomial gives a lamination for the H´ non map. e

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**Date Issued**

2013-08-19#####
**Subject**

Dynamical Systems; complex Hénon maps; semi-parabolic Hénon maps; laminations; structural stability; topological models

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**Committee Chair**

Hubbard, John Hamal

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**Committee Member**

Kozen, Dexter Campbell; Smillie, John D; Guckenheimer, John Mark

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**Degree Discipline**

Mathematics

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**Degree Name**

Ph.D. of Mathematics

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**Degree Level**

Doctor of Philosophy

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**Type**

dissertation or thesis