A structure theorem for semi-parabolic Hénon maps

Abstract

Consider the parameter space ${\mathcal{P}}_{\lambda }\subset {\mathbb{C}}^2$ of complex Hénon maps $$H_{c,a}(x,y)=(x^2+c+ay,ax),a\ne 0$$ which have a semi-parabolic fixed point with one eigenvalue $\lambda =e^{2\pi ip/q}$. We give a characterization of those Hénon maps from the curve ${\mathcal{P}}_{\lambda }$ that are small perturbations of a quadratic polynomial $p$ with a parabolic fixed point of multiplier $\lambda$. We prove that there is an open disk of parameters in ${\mathcal{P}}_{\lambda }$ for which the semi-parabolic Hénon map has connected Julia set $J$ and is structurally stable on $J$ and $J^{+}$. The Julia set $J^{+}$ has a nice local description: inside a bidisk ${\mathbb{D}}_r\times {\mathbb{D}}_r$ it is a trivial fiber bundle over $J_p$, the Julia set of the polynomial $p$, with fibers biholomorphic to ${\mathbb{D}}_r$. The Julia set $J$ is homeomorphic to a quotiented solenoid.

54 pages, incl. references; 8 figures

References in corpus (1)

• Parabolic Bifurcations in Complex Dimension 2

Cited by in corpus (3)

• Complex Hénon maps and discrete groups
• Structure of partially hyperbolic Hénon maps
• Dynamics of Polynomial Diffeomorphisms of $C^2$: Foliations and laminations