On g-functions for countable state subshifts

TL;DR

This paper extends the understanding of g-functions, which are related to the probability measures on subshifts, from finite to countable alphabet cases, providing necessary and sufficient conditions for their existence.

Contribution

It generalizes Krieger's results on finite alphabet subshifts to the more complex countable alphabet subshifts, establishing broader conditions for continuous g-functions.

Findings

Generalized conditions for the existence of continuous g-functions to countable alphabet subshifts

Extended Krieger's finite alphabet results to countable cases

Provided a theoretical framework for analyzing countable state subshifts

Abstract

This note revisits the problem of finding necessary and sufficient conditions for a subshift to have a continuous g-function. Results obtained by Krieger (IMS Lecture Notes-Monograph Series, 48, 306--316, 2006) on finite alphabet subshifts are generalized to countable state subshifts.