Isomorphism and embedding of Borel systems on full sets

TL;DR

This paper constructs unique universal Borel systems for each entropy level, classifies various dynamical systems by entropy up to null sets, and answers a question about entropy conjugacy.

Contribution

It introduces strictly t-universal Borel systems for each entropy level and classifies several classes of systems by entropy up to null sets.

Findings

Existence of unique strictly t-universal Borel systems for each t.

Classification of several dynamical systems by entropy up to null sets.

Any two systems with the same entropy in these classes are entropy conjugate.

Abstract

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy at least t by removing the periodic points and "restricting" to the part of the system of entropy <t. As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.