HéNon Maps, Discrete Groups And Continuity Of Julia Sets

Other Titles


Consider the family of complex H´ non maps H ( x, y) = ( p( x) [-] ay, x), where p e is a quadratic polynomial and a is a complex parameter. Let D be the open complex unit disk and U + the set of points that escape to infinity under forward iterations of the H´ non map. The analytic structure of the escaping set U + is well e understood from previous work of Hubbard and Oberste-Vorth as a quotient of (C [-] D) x C by a discrete group of automorphisms [GAMMA] isomorphic to Z[1/2]/Z. However, the boundary J + of U + is a complicated fractal object, on which the H´ non map behaves chaotically. e We show how to extend the group action to S1 x C, in order to represent J + as a quotient of S1 x C/[GAMMA] by an equivalence relation. We analyze this extension for H´ non maps that are small perturbations of hyperbolic polynomials with e connected Julia sets. The proof uses results of Lyubich and Robertson about the critical locus for H´ non maps and it requires a careful analysis of the invariants of the H´ non e e map. We discovered an invariant function of the H´ non map, which we called e [alpha]. We developed and implemented an algorithm for computing [alpha]. We studied the degeneracy of [alpha] as the Jacobian tends to 0, and came up with an interesting relation connecting the function [alpha] with the group action on S1 x C. H´ non maps with a semi-parabolic fixed point, and their perturbations in C2 , e play an important role in understanding the parameter space of complex H´ non e maps. A fixed point is semi-parabolic if the derivative of the H´ non map at that e fixed point has one eigenvalue [lamda] = e2[pi]ip/q and one eigenvalue smaller than one in absolute value. Semi-parabolic implosion in C2 has been recently studied by Bedford, Smillie and Ueda. They show how to perturb from semi-parabolic H´ non maps such that J does not move continuously with the parameters. e Let [lamda] be a root of unity. For t [GREATER-THAN OR EQUAL TO] 0 real and sufficiently small, we look at the parameter space P(1+t)[lamda] of complex H´ non maps which have a fixed point with e one eigenvalue (1 + t)[lamda]. When t = 0 these are semi-parabolic. We prove that the parametric region {(c, a) ∈ P[lamda] : 0 < |a| < [delta]} of semi-parabolic H´ non maps lies e in the boundary of a hyperbolic component of the H´ non connectedness locus. e We show that for (c, a) ∈ P(1+t)[lamda] and 0 < |a| < [delta], the Julia sets J and J + depend continuously on the parameters as t [RIGHTWARDS ARROW] 0+ .

Journal / Series

Volume & Issue



Date Issued




Dynamical Systems; Complex Hénon maps; Discrete groups; Semi-parabolic germs; Continuity of Julia sets; Hyperbolicity


Effective Date

Expiration Date




Union Local


Number of Workers

Committee Chair

Hubbard, John Hamal

Committee Co-Chair

Committee Member

Guckenheimer, John Mark
Gehrke, Johannes E.
Smillie, John D

Degree Discipline


Degree Name

Ph. D., Mathematics

Degree Level

Doctor of Philosophy

Related Version

Related DOI

Related To

Related Part

Based on Related Item

Has Other Format(s)

Part of Related Item

Related To

Related Publication(s)

Link(s) to Related Publication(s)


Link(s) to Reference(s)

Previously Published As

Government Document




Other Identifiers


Rights URI


dissertation or thesis

Accessibility Feature

Accessibility Hazard

Accessibility Summary

Link(s) to Catalog Record