Poisson suspensions and entropy for infinite transformations

TL;DR

This paper explores the concept of Poisson entropy for infinite-measure-preserving transformations, establishing its equivalence with other entropy definitions for certain classes and analyzing its properties and implications for factors and Pinsker factors.

Contribution

It proves the equivalence of Poisson entropy with Krengel's and Parry's entropy for quasi-finite transformations and establishes the existence of Pinsker factors with zero Poisson entropy.

Findings

Poisson entropy coincides with Krengel's and Parry's entropy for quasi-finite transformations

Poisson entropy dominates Parry's entropy in conservative transformations

Existence of a Pinsker factor with zero Poisson entropy is established

Abstract

The Poisson entropy of an infinite-measure-preserving transformation is defined as the Kolmogorov entropy of its Poisson suspension. In this article, we relate Poisson entropy with other definitions of entropy for infinite transformations: For quasi-finite transformations we prove that Poisson entropy coincides with Krengel's and Parry's entropy. In particular, this implies that for null-recurrent Markov chains, the usual formula for the entropy $-\sum q_i p_{i,j}\log p_{i,j}$ holds in any of the definitions for entropy. Poisson entropy dominates Parry's entropy in any conservative transformation. We also prove that relative entropy (in the sense of Danilenko and Rudolph) coincides with the relative Poisson entropy. Thus, for any factor of a conservative transformation, difference of the Krengel's entropy is equal to the difference of the Poisson entropies. In case there exists a factor with zero Poisson entropy, we prove the existence of a maximum (Pinsker) factor with zero Poisson entropy. Together with the preceding results, this answers affirmatively the question raised in arXiv:0705.2148v3 about existence of a Pinsker factor in the sense of Krengel for quasi-finite transformations.