Decomposition of a symbolic element over a countable amenable group into blocks approximating ergodic measures

Abstract

Consider a subshift over a finite alphabet, $X\subset {\Lambda }^{\mathbb{Z}}$ (or $X\subset {\Lambda }^{{\mathbb{N}}_0}$). With each finite block $B\in {\Lambda }^k$ appearing in $X$ we associate the empirical measure ascribing to every block $C\in {\Lambda }^l$ the frequency of occurrences of $C$ in $B$. By comparing the values ascribed to blocks $C$ we define a metric on the combined space of blocks $B$ and probability measures $\mu$ on $X$, whose restriction to the space of measures is compatible with the weak-$\star$ topology. Next, in this combined metric space we fix an open set $\mathcal{U}$ containing all ergodic measures, and we say that a block $B$ is “ergodic” if $B\in \mathcal{U}$. In this paper we prove the following main result: Given $\epsilon >0$, every $x\in X$ decomposes as a concatenation of blocks of bounded lengths in such a way that, after ignoring a set $M$ of coordinates of upper Banach density smaller than $\epsilon$, all blocks in the decomposition are ergodic. We also prove a finitistic version of this theorem (about decomposition of long blocks), and a version about decomposition of $x\in X$ into finite blocks of unbounded lengths. The second main result concerns subshifts whose set of ergodic measures is closed. We show that, in this case, no matter how $x\in X$ is partitioned into blocks (as long as their lengths are sufficiently large and bounded), excluding a set $M$ of upper Banach density smaller than $\epsilon$, all blocks in the decomposition are ergodic. The first half of the paper is concluded by examples showing, among other things, that the small set $M$, in both main theorems, cannot be avoided. The second half of the paper is devoted to generalizing the two main results described above to subshifts $X\subset {\Lambda }^G$ with the action of a countable amenable group $G$. The role of long blocks is played by blocks whose domains are members of a Følner sequence, while the decomposition of $x\in X$ into blocks (of which majority are ergodic) is obtained with the help of a congruent system of tilings.