Balancedness and coboundaries in symbolic systems

TL;DR

This paper explores balancedness in infinite words and subshifts, focusing on dendric and substitutive words, and establishes criteria for balancedness and the role of coboundaries in these systems.

Contribution

It provides new characterizations of balancedness for dendric and substitutive words, highlighting the significance of coboundaries with rational values.

Findings

Dendric words are balanced on letters iff balanced on factors.

Existence of rational coboundaries influences imbalancedness.

Criteria for rational frequency values and their impact on balancedness.

Abstract

This paper studies balancedness for infinite words and subshifts, both for letters and factors. Balancedness is a measure of disorder that amounts to strong convergence properties for frequencies. It measures the difference between the numbers of occurrences of a given word in factors of the same length. We focus on two families of words, namely dendric words and words generated by substitutions. The family of dendric words includes Sturmian and Arnoux-Rauzy words, as well as codings of regular interval exchanges. We prove that dendric words are balanced on letters if and only if they are balanced on words. In the substitutive case, we stress the role played by the existence of coboundaries taking rational values and show simple criteria when frequencies take rational values for exhibiting imbalancedness.