Sinai factors of nonsingular systems: Bernoulli shifts and Anosov flows

TL;DR

This paper extends Sinai's factor theorem to nonsingular systems, showing limitations on finitary factors of Bernoulli shifts and establishing iid factors for Anosov diffeomorphisms with entropy constraints.

Contribution

It proves that nonsingular Bernoulli shifts can have iid finitary factors with entropy below a certain threshold, extending Sinai's theorem to the nonsingular setting.

Findings

Finitary factors of Bernoulli shifts are limited by entropy constraints.

Every transitive Anosov diffeomorphism has iid factors with entropy bounds.

Extension of Sinai's factor theorem to nonsingular systems.

Abstract

We show that a totally dissipative system has all nonsingular systems as factors, but that this is no longer true when the factor maps are required to be finitary. In particular, if a nonsingular Bernoulli shift satisfies the Doeblin condition, and has a limiting marginal distribution p, then it cannot have, as a finitary factor, an independent and identically distributed (iid) system of entropy larger than H(p); on the other hand, we show that iid systems with entropy strictly lower than H(p) can be obtained as finitary factors of these Bernoulli shifts, extending Keane and Smorodinsky's finitary version of Sinai's factor theorem to the nonsingular setting. As a consequence of our results we also obtain that every transitive twice continuously differentiable Anosov diffeomorphism on a compact manifold, endowed with volume measure, has iid factors, and if the factor is required to be finitary, then the iid factor cannot have Kolmogorov-Sinai entropy greater than the measure-theoretic entropy of a Sinai-Ruelle-Bowen measure associated with the Anosov diffeomorphism.