# Measure concentration and the weak Pinsker property

## TL;DR

This paper proves that all ergodic automorphisms have the weak Pinsker property, showing they can be approximated by Bernoulli shifts plus low-entropy automorphisms, using new measure concentration results.

## Contribution

It establishes the weak Pinsker property for all ergodic automorphisms and introduces a general measure concentration theorem for product measures.

## Key findings

- All ergodic automorphisms have the weak Pinsker property.
- Any factor map can be enlarged by little entropy to become relatively Bernoulli.
- Probability measures on finite product spaces can be decomposed into concentrated measures with bounded complexity.

## Abstract

Let $(X,\mu)$ be a standard probability space. An automorphism $T$ of $(X,\mu)$ has the weak Pinsker property if for every $\varepsilon > 0$ it has a splitting into a direct product of a Bernoulli shift and an automorphism of entropy less than $\varepsilon$. This property was introduced by Thouvenot, who asked whether it holds for all ergodic automorphisms. This paper proves that it does. The proof actually gives a more general result. Firstly, it gives a relative version: any factor map from one ergodic automorphism to another can be enlarged by arbitrarily little entropy to become relatively Bernoulli. Secondly, using some facts about relative orbit equivalence, the analogous result holds for all free and ergodic measure-preserving actions of a countable amenable group. The key to this work is a new result about measure concentration. Suppose now that $\mu$ is a probability measure on a finite product space $A^n$, and endow this space with its Hamming metric. We prove that $\mu$ may be represented as a mixture of other measures in which (i) most of the weight in the mixture is on measures that exhibit a strong kind of concentration, and (ii) the number of summands is bounded in terms of the difference between the Shannon entropy of $\mu$ and the combined Shannon entropies of its marginals.