The solenoid and holomorphic motions for Henon maps

Holomorphic Shadowing for Hénon Maps Revisited: an Implicit Function Theorem Perspective

In studying the complex Hénon maps, Mummert (in "Holomorphic shadowing for Hénon maps" Nonlinearity 21 pp. 2887-2898, 2008) defined an operator the fixed points of which give rise to bounded orbits. This enabled him to obtain an estimate of the solenoid locus. Instead of the contraction mapping theorem, in this paper we present an implicit function theorem version of his result by providing an alternative proof of a theorem of Hubbard and Oberste-Vorth (in Real and Complex Dynamical Systems, pp.89-132, 1995).

$λ$-Lemma for families of Riemann surfaces and the critical loci of complex Hénon maps

We prove a version of the classical $λ$-lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex Hénon maps that are small perturbations of quadratic polynomials with Cantor Julia sets are all quasiconformally equivalent.

A structure theorem for semi-parabolic Hénon maps

Consider the parameter space $\mathcal{P}_λ\subset \mathbb{C}^{2}$ of complex Hénon maps $$ H_{c,a}(x,y)=(x^{2}+c+ay,ax),\ \ a\neq 0 $$ which have a semi-parabolic fixed point with one eigenvalue $λ=e^{2πi p/q}$. We give a characterization of those Hénon maps from the curve $\mathcal{P}_λ$ that are small perturbations of a quadratic polynomial $p$ with a parabolic fixed point of multiplier $λ$. We prove that there is an open disk of parameters in $\mathcal{P}_λ$ for which the semi-parabolic Hénon map has connected Julia set $J$ and is structurally stable on $J$ and $J^{+}$. The Julia set $J^{+…

关于某些高秩自由群的全纯与随机动力学的示例

On the Holomorphic and Random Dynamics for some examples of higher rank Free Groups generated by Hénon type maps

本文研究了由Hénon型映射生成的高秩自由群的全纯与随机动力学，证明了Fatou集非空以及平稳测度的存在性，并构造了无平稳测度的例子，揭示了非紧流形上全纯群作用的典型现象。

A compactification of Henon mappings in C^2 as dynamical systems

In \cite {HO1}, it was shown that there is a topology on $\C^2\sqcup S^3$ homeomorphic to a 4-ball such that the Hénon mapping extends continuously. That paper used a delicate analysis of some asymptotic expansions, for instance, to understand the structure of forward images of lines near infinity. The computations were quite difficult, and it is not clear how to generalize them to other rational maps. In this paper we will present an alternative approach, involving blow-ups rather than asymptotics. We apply it here only to Hénon mappings and their compositions, but the method should work quit…

Dynamical properties of families of holomorphic mappings

We study some dynamical properties of skew products of Hénon maps of $\mbb C^2$ that are fibered over a compact metric space $M$. The problem reduces to understanding the dynamical behavior of the composition of a pseudo-random sequence of Hénon mappings. In analogy with the dynamics of the iterates of a single Hénon map, it is possible to construct fibered Green's functions that satisfy suitable invariance properties and the corresponding stable and unstable currents. This analogy is carried forth in two ways: it is shown that the successive pullbacks of a suitable current by the skew Hénon m…