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

[摘要] Consider a subshift over a finite alphabet, X⊂ΛZX\subset \Lambda^\mathbb{Z}X⊂ΛZ (or X⊂ΛN0X\subset\Lambda^{\mathbb{N}_0}X⊂ΛN0​). With each finite block B∈ΛkB\in\Lambda^kB∈Λk appearing in XXX we associate the empirical measure ascribing to every block C∈ΛlC\in\Lambda^lC∈Λl the frequency of occurrences of CCC in BBB. By comparing the values ascribed to blocks CCC we define a metric on the combined space of blocks BBB and probability measures μ\muμ on XXX, 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 U\mathcal{U}U containing all ergodic measures, and we say that a block BBB is “ergodic” if B∈UB\in\mathcal{U}B∈U. In this paper we prove the following main result: Given ε00ε0, every x∈Xx\in Xx∈X decomposes as a concatenation of blocks of bounded lengths in such a way that, after ignoring a set MMM of coordinates of upper Banach density smaller than ε\varepsilonε, 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∈Xx\in Xx∈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∈Xx\in Xx∈X is partitioned into blocks (as long as their lengths are sufficiently large and bounded), excluding a set MMM of upper Banach density smaller than ε\varepsilonε, 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 MMM, 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⊂ΛGX\subset\Lambda^GX⊂ΛG with the action of a countable amenable group GGG. The role of long blocks is played by blocks whose domains are members of a Følner sequence, while the decomposition of x∈Xx\in Xx∈X into blocks (of which majority are ergodic) is obtained with the help of a congruent system of tilings.

[学科分类] 神经科学

[关键词] Symbolic systems;ergodic decomposition;infinite concatenation;amenable group action;tilings