# HéNon Maps, Discrete Groups And Continuity Of Julia Sets

dc.contributor.author | Tanase, Raluca | en_US |

dc.contributor.chair | Hubbard, John Hamal | en_US |

dc.contributor.committeeMember | Guckenheimer, John Mark | en_US |

dc.contributor.committeeMember | Gehrke, Johannes E. | en_US |

dc.contributor.committeeMember | Smillie, John D | en_US |

dc.date.accessioned | 2013-09-16T16:43:09Z | |

dc.date.available | 2018-08-20T06:01:19Z | |

dc.date.issued | 2013-08-19 | en_US |

dc.description.abstract | 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+ . | en_US |

dc.identifier.other | bibid: 8267452 | |

dc.identifier.uri | https://hdl.handle.net/1813/34340 | |

dc.language.iso | en_US | en_US |

dc.subject | Dynamical Systems | en_US |

dc.subject | Complex Hénon maps | en_US |

dc.subject | Discrete groups | en_US |

dc.subject | Semi-parabolic germs | en_US |

dc.subject | Continuity of Julia sets | en_US |

dc.subject | Hyperbolicity | en_US |

dc.title | HéNon Maps, Discrete Groups And Continuity Of Julia Sets | en_US |

dc.type | dissertation or thesis | en_US |

thesis.degree.discipline | Mathematics | |

thesis.degree.grantor | Cornell University | en_US |

thesis.degree.level | Doctor of Philosophy | |

thesis.degree.name | Ph. D., Mathematics |

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