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Semi-parabolic tools for hyperbolic Hénon maps and continuity of Julia sets in $ \mathbb{C}^{2}$


Authors: Remus Radu and Raluca Tanase
Journal: Trans. Amer. Math. Soc. 370 (2018), 3949-3996
MSC (2010): Primary 37F45, 37D99, 32A99, 47H10
DOI: https://doi.org/10.1090/tran/7061
Published electronically: December 18, 2017
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Abstract: We prove some new continuity results for the Julia sets $ J$ and $ J^{+}$ of the complex Hénon map $ H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $ a$ and $ c$ are complex parameters. We look at the parameter space of dissipative Hénon maps which have a fixed point with one eigenvalue $ (1+t)\lambda $, where $ \lambda $ is a root of unity and $ t$ is real and small in absolute value. These maps have a semi-parabolic fixed point when $ t$ is 0, and we use the techniques that we have developed in a prior work for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero $ \vert t\vert$, the Hénon map is hyperbolic and has connected Julia set. We prove that the Julia sets $ J$ and $ J^{+}$ depend continuously on the parameters as $ t\rightarrow 0$, which is a two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of Hénon maps is stable on $ J$ and $ J^{+}$ when $ t$ is non-negative.


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Additional Information

Remus Radu
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660
Email: remus.radu@stonybrook.edu

Raluca Tanase
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660
Email: raluca.tanase@stonybrook.edu

DOI: https://doi.org/10.1090/tran/7061
Received by editor(s): September 3, 2015
Received by editor(s) in revised form: July 22, 2016, and September 6, 2016
Published electronically: December 18, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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