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

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. 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.