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

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- by Remus Radu and Raluca Tanase PDF
- Trans. Amer. Math. Soc.
**370**(2018), 3949-3996 Request permission

## 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 $|t|$, 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.## References

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## Additional Information

**Remus Radu**- Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660
- MR Author ID: 1156737
- Email: remus.radu@stonybrook.edu
**Raluca Tanase**- Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660
- MR Author ID: 1156840
- Email: raluca.tanase@stonybrook.edu
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 3949-3996 - MSC (2010): Primary 37F45, 37D99, 32A99, 47H10
- DOI: https://doi.org/10.1090/tran/7061
- MathSciNet review: 3811516