Every Borel automorphism without finite invariant measures admits a two-set generator

TL;DR

This paper proves that Borel automorphisms without finite invariant measures have a simple two-set generator, leading to entropy-based classification results for hyperbolic-like systems at the Borel level.

Contribution

It establishes the existence of a two-set generator for automorphisms lacking finite invariant measures and derives entropy-based classification results for certain Borel systems.

Findings

Automorphisms without finite invariant measures admit a two-set generator.

Systems with invariant measure entropies less than log(k) have a k-set generator.

Hyperbolic-like systems are classified by entropy and periodic points at the Borel level.

Abstract

We show that if an automorphism of a standard Borel space does not admit finite invariant measures, then it has a two-set generator modulo the sigma-ideal generated by wandering sets. This implies that if the entropies of invariant probability measures of a Borel system are all less than log(k), then the system admits a k-set generator, and that a wide class of hyperbolic-like systems are classified completely at the Borel level by entropy and periodic points counts.