Math page of Thierry Monteil

Blog -- Unique ergodicity is necessary to ensure frequencies

Introduction

The first motivating question was the following : we are looking for a condition on the complexity of an infinite word that ensures that it has frequencies. We are mostly interested with uniformly recurrent words.

Michael Boshernitzan proved that if a uniformly recurrent word $x$ satisfies $\limsup \frac{p_n(x)}{n}<3$ or $\liminf \frac{p_n(x)}{n}< 2$ then its associated subshift is uniquely ergodic, hence $x$ has frequencies [1]. Julien Cassaigne and Idrissa Kaboré constructed a word $x$ without frequencies such that $(\forall n \in \mathbb{N}^*) ( p_n(x) < 3n)$.

Problem

More generally, suppose we are looking for sufficient conditions on the language of an uniformly recurrent word to have frequencies.

Looking for the unique ergodicity of the associated subshift is a possible strategy (see Background - Symbolic dynamics), but we may miss something (it is a sufficient condition). In fact, unique ergodicity is necessary to ensure that all words sharing a uniformly recurrent language have frequencies.

Proof

The proof comes from an old paper of Oxtoby [2] : Suppose that there exists a minimal non-uniquely ergodic subshift $X$ such that any word in $X$ has frequencies. Any word $w$ in $X$ has frequencies so the functions $$f_w(x):=\lim_{n\rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n-1} \chi_{[w]}(x) $$ are well defined on $X$. Since $X$ is not uniquely ergodic, there exists two ergodic measures $\mu\neq\nu$, hence there is a finite word $w$ such that $\mu[w]\neq\nu[w]$. Let $x$ (resp. $y$) be a generic point for $\mu$ (resp. $\nu$). We have $f_w(x)=\mu[w]\neq\nu[w]=f(y)$ and and since $f_w$ is constant along the orbits, $f_w$ is nowhere continuous (the minimality implies that the orbits of $x$ and $y$ are dense). But $f_w$ is a simple limit of continuous functions on a complete metric space ($X$ is compact), hence Baire's lemma implies that the points of continuity of $f_w$ must be dense in $X$. A contradiction.

Hence looking for unique ergodicity is not just a trick and there is no loss to look for it.

References

1. M. Boshernitzan, A unique ergodicity of minimal symbolic flows with linear block growth, J. Analyse Math. 44 (1984/85), p 77-96.
2. J.C. Oxtoby, Ergodic sets, Bull. Amer. Math. Soc. , 58 (1952) pp. 116-136.