Follower, Predecessor, and Extender Entropies

TL;DR

This paper introduces follower, predecessor, and extender entropies for shift spaces, analyzing their invariance properties and demonstrating their usefulness in distinguishing non-isomorphic shift spaces.

Contribution

It defines new entropy measures for shift spaces and proves invariance properties, providing tools to distinguish shift spaces beyond classical entropy.

Findings

Extender entropy is a conjugacy invariant.

Having follower entropy zero is a conjugacy invariant.

Examples show these entropies can distinguish non-isomorphic shift spaces.

Abstract

Using follower/predecessor/extender set sequences, we define quantities which we call the follower/predecessor/extender entropies, which can be associated to any shift space. We analyze the behavior of these quantities under conjugacies and factor maps, most notably showing that extender entropy is a conjugacy invariant and that having follower entropy zero is a conjugacy invariant. We give some applications, including examples of shift spaces with equal entropy which can be distinguished by extender entropy, and examples of shift spaces which can be shown to not be isomorphic to their inverse by using follower/predecessor entropy.