Two results on entropy, chaos, and independence in symbolic dynamics

TL;DR

This paper explores fundamental links between entropy, chaos, and independence in symbolic dynamics, providing new proofs and extending classical results to better understand the structure of shift spaces.

Contribution

It introduces new proofs and extends classical results relating entropy, chaos, and independence in symbolic dynamics, emphasizing the structure of shift spaces.

Findings

Positive entropy corresponds to large independence sets in shift spaces.

Existence of a mixing shift space with zero entropy, dense periodic points, and no ergodic measure with full support.

New proofs yield stronger conclusions than previous results.

Abstract

We survey the connections between entropy, chaos, and independence in topological dynamics. We present extensions of two classical results placing the following notions in the context of symbolic dynamics: 1. Equivalence of positive entropy and the existence of a large (in terms of asymptotic and Shnirelman densities) set of combinatorial independence for shift spaces. 2. Existence of a mixing shift space with a dense set of periodic points with topological entropy zero and without ergodic measure with full support, nor any distributionally chaotic pair. Our proofs are new and yield conclusions stronger than what was known before.