Topological entropy of the set of generic points for $(\alpha-\beta)$-shifts

TL;DR

This paper proves that for a class of symbolic dynamical systems called ($ ext{α}$-$ ext{β}$)-shifts with specific parameters, the topological entropy of generic points matches the measure-theoretic entropy, demonstrating a saturation property.

Contribution

It establishes the saturation property for all ($ ext{α}$-$ ext{β}$)-shifts with $0 \\leq \\alpha < 1$ and $eta > 2$, a significant extension in the understanding of their entropy characteristics.

Findings

All ($ ext{α}$-$ ext{β}$)-shifts with specified parameters are saturated.

The topological entropy of generic points equals the measure-theoretic entropy.

The result applies to a broad class of symbolic dynamical systems.

Abstract

We prove that all ($\alpha$-$\beta$)-shifts with $0\le \alpha<1$ and $\beta>2$ are saturated, that is, for any invariant measure, the topological entropy of the set of generic points coincides with the metric entropy.