Criteria for Unique Ergodicity of Symbolic Dynamical Systems

Abstract

Boshernitzan proved that under certain growth conditions on the complexity function of minimal symbolic dynamical systems, there exist upper bounds for the number of ergodic measures. The proofs of these criteria are due to the ubiquity of special words in sequences of low complexity. Motivated by the goal of sharpening Boshernitzan’s bounds, we define and study the special Rauzy graph, which is a variant of the Rauzy graph that focuses on these special words. We establish properties of the special Rauzy graphs and explicitly compute them for the Morse sequence and related sequences. Finally, we describe how these graphs relate to Boshernitzan’s criteria and show how they can be used to bound the number of ergodic measures of a symbolic dynamical system.