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

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 [RT] 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 $t\to 0$, 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 nonnegative.

Final version, to appear in Transactions of the AMS; continues arXiv:1411.3824

References in corpus (4)

• Henon mappings in the complex domain II: projective and inductive limits of polynomials
• A structure theorem for semi-parabolic Hénon maps
• Parabolic Bifurcations in Complex Dimension 2
• On parameter loci of the Hénon family

Cited by in corpus (1)

• A new proof of a theorem of Hubbard-Oberste-Vorth