Non-uniform hyperbolicity in polynomial skew products

TL;DR

This paper investigates the dynamics of polynomial skew products in complex two-dimensional space, showing that under certain non-uniform hyperbolicity conditions, the Fatou set simplifies and the Julia set has measure zero, eliminating wandering components.

Contribution

It establishes new results on the structure of Fatou and Julia sets for polynomial skew products under non-uniform hyperbolicity assumptions.

Findings

Fatou set in basin of L equals union of attracting cycle basins

Julia set in basin of L has Lebesgue measure zero

No wandering Fatou components in the basin of L

Abstract

Let $f:\mathbb{C}^2\to \mathbb{C}^2$ be a polynomial skew product which leaves invariant an attracting vertical line $ L $. Assume moreover $f$ restricted to $L$ is non-uniformly hyperbolic, in the sense that $f$ restricted to $L$ satisfies one of the following conditions: 1. $f|_L$ satisfies Topological Collet-Eckmann and Weak Regularity conditions. 2. The Lyapunov exponent at every critical value point lying in the Julia set of $f|_L$ exist and is positive, and there is no parabolic cycle. Under one of the above conditions we show that the Fatou set in the basin of $L$ coincides with the union of the basins of attracting cycles, and the Julia set in the basin of $L$ has Lebesgue measure zero. As an easy consequence there are no wandering Fatou components in the basin of $L$.