Introduction to weak Pinsker filtrations
Séverin Benzoni
TL;DR
This work analyzes weak Pinsker filtrations, filtrations of a dynamical system whose successive factors are relatively Bernoulli and whose entropy vanishes along the tail, linking them to the Pinsker factor. Building on Austin's result that every ergodic system has the weak Pinsker property, the authors develop a dynamical-filtration framework, study uniqueness questions, and prove a partial Bernoulli-uniqueness theorem: for finite-entropy Bernoulli systems there exists a subsequence of a weak Pinsker filtration that is of product type. They also construct explicit examples via cellular automata on both Bernoulli shifts and Ornstein's non-Bernoulli K-process to illustrate how weak Pinsker filtrations can arise and differ in structure depending on the underlying system. These results advance a potential classification tool for positive-entropy systems, highlighting fundamental differences between Bernoulli and non-Bernoulli K-systems and suggesting avenues for a deeper understanding of dynamical filtrations in ergodic theory.
Abstract
We introduce the so-called weak Pinsker dynamical filtrations, whose existence in any ergodic system follows from the universality of the weak Pinsker property, recently proved by Austin. These dynamical filtrations appear as a potential tool to describe and classify positive entropy systems. We explore the links between the asymptotic structure of weak Pinsker filtrations and the properties of the underlying dynamical system. A central question is whether, on a given system, the structure of weak Pinsker filtrations is unique up to isomorphism. We give a partial answer, in the case where the underlying system is Bernoulli. We conclude our work by giving two explicit examples of weak Pinsker filtrations.