Local homeomorphisms that *-commute with the shift

TL;DR

This paper explores the conditions under which sliding block codes on shift spaces are local homeomorphisms and *-commute with the shift, introducing new concepts like weakly progressive and regressive block maps to characterize these properties.

Contribution

It generalizes the notion of progressive block maps to weakly progressive ones and characterizes *-commuting sliding block codes via regressive block maps, providing new insights into their structure.

Findings

A counterexample shows the converse of Exel and Renault's result does not hold.

Sliding block codes from weakly progressive maps are local homeomorphisms.

Sliding block codes that *-commute with the shift are characterized by regressive block maps.

Abstract

Exel and Renault proved that a sliding block code on a one-sided shift space coming from a progressive block map is a local homeomorphism. We provide a counterexample showing that the converse does not hold. We use this example to generalize the notion of progressive to a property of block maps we call weakly progressive, and we prove that a sliding block code coming from a weakly progressive block map is a local homeomorphism. We also introduce the notion of a regressive block map and prove that a sliding block code *-commutes with the shift map if and only if it comes from a regressive block map. We also prove that a sliding block code is a local homeomorphism and *-commutes with the shift map if and only if it is a k-fold covering map defined from a regressive block map.