Topologically completely positive entropy and zero-dimensional topologically completely positive entropy

TL;DR

This paper explores the concept of zero-dimensional topologically completely positive entropy (ZTCPE) in $ abla$-shifts of finite type, comparing it with classical TCPE, and provides conditions under which ZTCPE implies TCPE.

Contribution

It introduces the distinction between ZTCPE and TCPE, showing examples where ZTCPE does not imply TCPE, and offers a strengthened condition that guarantees TCPE for $ abla$-SFTs.

Findings

Existence of subshifts with ZTCPE but not TCPE in 1D and 2D.

A sufficient condition for $ abla$-SFTs to have TCPE.

Clarification of the relationship between ZTCPE and TCPE.

Abstract

In a previous paper ("A characterization of topologically completely positive entropy for shifts of finite type"), the author gave a characterization for when a $\mathbb{Z}^d$-shift of finite type (SFT) has no nontrivial subshift factors with zero entropy, a property which we here call zero-dimensional topologically completely positive entropy (ZTCPE). In this work, we study the difference between this notion and the more classical topologically completely positive entropy (TCPE) of Blanchard. We show that there are one-dimensional subshifts and two-dimensional SFTs which have ZTCPE but not TCPE. In addition, we show that strengthening the hypotheses of the main result of the aforementioned paper yields a sufficient condition for a $\mathbb{Z}^d$-SFT to have TCPE.