Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts

TL;DR

This paper characterizes positive topological entropy and distributional chaos in hereditary shifts, including spacing and beta shifts, and solves open problems by linking entropy with recurrence properties and chaos types.

Contribution

It provides a complete characterization of chaos and entropy in hereditary shifts, applying these results to spacing and beta shifts, and introduces new links with recurrence and difference sets.

Findings

Hereditary shift has positive entropy iff it is DC2- or DC3-chaotic.

Spacing shift has positive entropy iff its complement is Poincaré recurrent.

Constructed a proximal spacing shift with positive entropy.

Abstract

Positive topological entropy and distributional chaos are characterized for hereditary shifts. A hereditary shift has positive topological entropy if and only if it is DC2-chaotic (or equivalently, DC3-chaotic) if and only if it is not uniquely ergodic. A hereditary shift is DC1-chaotic if and only if it is not proximal (has more than one minimal set). As every spacing shift and every beta shift is hereditary the results apply to those classes of shifts. Two open problems on topological entropy and distributional chaos of spacing shifts from an article of Banks et al. are solved thanks to this characterization. Moreover, it is shown that a spacing shift $\Omega_P$ has positive topological entropy if and only if $\mathbb{N}\setminus P$ is a set of Poincar\'{e} recurrence. Using a result of K\v{r}\'{\i}\v{z} an example of a proximal spacing shift with positive entropy is constructed. Connections between spacing shifts and difference sets are revealed and the methods of this paper are used to obtain new proofs of some results on difference sets.