A hierarchy of topological systems with completely positive entropy

TL;DR

This paper introduces a hierarchical classification of topological systems with completely positive entropy for countable amenable group actions, constructing examples at each level and linking entropy pairs to asymptotic relations.

Contribution

It develops a hierarchy of systems with completely positive entropy, constructs explicit examples at each level, and connects entropy pairs with asymptotic relations, answering an open question.

Findings

Constructed systems for each ordinal level of the hierarchy.

Provided subshifts of finite type for the first three levels.

Established necessary and sufficient conditions for entropy pairs.

Abstract

We define a hierarchy of systems with topological completely positive entropy in the context of continuous countable amenable group actions on compact metric spaces. For each countable ordinal we construct a dynamical system on the corresponding level of the aforementioned hierarchy and provide subshifts of finite type for the first three levels. We give necessary and sufficient conditions for entropy pairs by means of the asymptotic relation on systems with the pseudo-orbit tracing property, and thus create a bridge between a result by Pavlov and a result by Meyerovitch. As a corollary, we answer negatively an open question by Pavlov regarding necessary conditions for completely positive entropy.