A closing lemma for polynomial automorphisms of C^2

TL;DR

This paper proves that for most polynomial automorphisms of complex two-dimensional space, invariant measures are supported on the closure of saddle periodic points, revealing a deep connection between invariant measures and periodic dynamics.

Contribution

It establishes a closing lemma for polynomial automorphisms of C^2, linking invariant measures to saddle periodic points, which was previously not fully understood.

Findings

Invariant measures are supported on the closure of saddle periodic points.

Supports of invariant measures are contained in the closure of saddle periodic points.

The result applies to all but a few trivial cases.

Abstract

We prove that for a polynomial diffeomorphism of C^2 , the support of any invariant measure, apart from a few obvious cases, is contained in the closure of the set of saddle periodic points.