Zero Krengel Entropy does not kill Poisson Entropy

TL;DR

This paper demonstrates that for infinite-measure-preserving transformations, zero Krengel entropy does not necessarily imply zero Poisson entropy, by constructing a specific counterexample.

Contribution

It provides the first explicit example showing the divergence between Krengel and Poisson entropy in infinite measure settings.

Findings

Constructed a conservative infinite-measure-preserving transformation with zero Krengel entropy.

Showed the associated Poisson suspension has positive entropy.

Established the non-equivalence of Krengel and Poisson entropy in certain cases.

Abstract

We prove that the notions of Krengel entropy and Poisson entropy for infinite-measure-preserving transformations do not always coincide: We construct a conservative infinite-measure-preserving transformation with zero Krengel entropy (the induced transformation on a set of measure 1 is the Von Neumann-Kakutani odometer), but whose associated Poisson suspension has positive entropy.