On entropy and intrinsic ergodicity of coded subshifts

TL;DR

This paper investigates the entropy and ergodic properties of coded subshifts, establishing conditions under which their entropy is determined by a simpler subshift and analyzing the uniqueness of measures of maximal entropy.

Contribution

It provides a new formula for the topological entropy of coded subshifts based on a limit subshift and a series condition, and shows when these systems have a unique measure of maximal entropy.

Findings

Entropy equals that of the limit subshift when a series condition holds.

A formula for entropy when the series exceeds 1.

Existence of a unique measure of maximal entropy in certain cases.

Abstract

Any coded subshift X defined by a set C of code words contains a subshift, which we call L, consisting of limits of single code words. We show that when C satisfies a unique decomposition property, the topological entropy h(X) of X is determined completely by h(L) and the number of code words of each length. More specifically, we show that h(X) = h(L) exactly when a certain infinite series is less than or equal to 1, and when that series is greater than 1, we give a formula for h(X_C). In the latter case, an immediate corollary (using results of Climenhaga and Thompson) is that X has a unique measure of maximal entropy.