Balancedness and coboundaries in symbolic systems

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.

On the dimension group of unimodular $${\mathcal {S}}$$-adic subshifts

“…Balance property is usually expressed for letters and factors (see for instance [6]). Indeed a minimal subshift (X, S) is said to be balanced on the factor v ∈ L(X) if χ [v] : X → {0, 1} is balanced, or, equivalently, if there exists a constant C v such that for all w, w in L X with |w| = |w |, then…”

“…It is conjugate to the shift generated by the substitution τ : a → bb, b → bd, c → ca, d → cb via the sliding block code 00 → a, 01 → b, 10 → c, 11 → d (see [62, p.149]). The subshift generated by τ is not balanced on letters (see [6]).…”

“…Note also that the triviality of Inf(K 0 (X, S)) says nothing about the balance property (see Example 6.4), but the existence of non-trivial infinitesimals indicates that some letter is not balanced. Lastly, the Thue-Morse substitution σ : a → ab, b → ba generates a subshift that is balanced on letters but not on factors [6]. This substitution is neither unimodular, nor proper.…”

“…Inspired by the classical notion of balance in word combinatorics (see e.g. references in [6]), we also introduce the notion of balanced functions and provide a topological characterization of this balance property for primitive unimodular proper S-adic subshifts (see Corollary 5.5).…”

Sturmian Ground States in Classical Lattice–Gas Models

“…Results of this type have long been studied under various names. As an example, for related results in more general symbolic systems, see [11], and classically [25] in the context of Diophantine approximation. For completeness we provide the straightforward proof.…”