Some rigidity results for polynomial automorphisms of C^2

TL;DR

This paper establishes new rigidity properties of polynomial automorphisms of ^2 with positive entropy, showing restrictions on Julia sets, foliations, and conjugacy, and analyzing multipliers over number fields.

Contribution

It introduces novel rigidity results for polynomial automorphisms of ^2, including constraints on Julia sets and conjugacy conditions.

Findings

A complex slice of the Julia set is never a smooth or rectifiable curve.

Such automorphisms cannot preserve a global holomorphic or real-analytic foliation with complex leaves.

Under mild conditions, real-analytically conjugate automorphisms are polynomially conjugate.

Abstract

We prove several new rigidity results for polynomial automorphisms of $\mathbb C^2$ with positive entropy. A first result is that a complex slice of the (forward or backward) Julia set is never a smooth, or even rectifiable, curve. We also show that such an automorphism cannot preserve a global holomorphic foliation, nor a real-analytic foliation with complex leaves. These results are used to show that under mild assumptions, two real-analytically conjugate automorphisms are polynomially conjugate. For mappings defined over a number field, we also study the fields of definition of multipliers of saddle periodic orbits.