Structure of hyperbolic polynomial automorphisms of C^2 with disconnected Julia sets

TL;DR

This paper provides a topological classification of the components of Julia sets for hyperbolic polynomial automorphisms of C^2 with disconnected Julia sets, revealing a finite structure governed by quasi-solenoids.

Contribution

It introduces a refined spectral decomposition for hyperbolic maps in two dimensions, extending one-dimensional polynomial dynamics theories.

Findings

Finitely many quasi-solenoids determine orbit behavior

Julia set components exhibit a John-like geometric property

Provides a topological description under mild dissipativity conditions

Abstract

For a hyperbolic polynomial automorphism of C^2 with a disconnected Julia set, and under a mild dissipativity condition, we give a topological description of the components of the Julia set. Namely, there are finitely many "quasi-solenoids" that govern the asymptotic behavior of the orbits of all non-trivial components. This can be viewed as a refined Spectral Decomposition for a hyperbolic map, as well as a two-dimensional version of the (generalized) Branner-Hubbard theory in one-dimensional polynomial dynamics. An important geometric ingredient of the theory is a John-like property of the Julia set in the unstable leaves.