论文标题
符号元素在可数符合的基团上分解成近似措施的块
Decomposition of a symbolic element over a countable amenable group into blocks approximating ergodic measures
论文作者
论文摘要
考虑通过有限字母的子换档，$ x \subsetλ^{\ mathbb z} $（或$ x \subsetλ^{\ mathbb n_0} $）。随着每个有限块$ b \inλ^k $出现在$ x $中，我们将归因于每个块$ c \inλ^l $归因于$ b $中的$ c $的频率。通过比较归因于块$ c $的值，我们在块$ b $的组合空间上定义了一个度量，概率是$ x $上的$μ$，$ x $的限制与弱的 - $ \ star $ topology兼容。接下来，在这个组合的度量空间中，我们修复了一个打开的套装$ \ MATHCAL U $，其中包含所有Ergodic措施，我们说如果$ b \ in \ Mathcal u $，则块$ b $是“ ergodic”。 在本文中，我们证明了以下主要结果：给定$ \ varepsilon> 0 $，x $中的每个$ x \分解为有界长度的块的串联，以至于忽略了$ \ \ varepsilon $的上班那上的$ m $ coordinate $ m $ coordinate compostip e ergodics。第二个主要结果涉及关闭一套千古措施的次班。我们表明，在这种情况下，无论x $中的$ x \ in x $如何分配到块中（只要它们的长度足够大且有界），在忽略了小于$ \ varepsilon $的上部$ m $的$ m $之后，分解中的所有块都是奇异的。本文的前半部分以示例得出结论，除其他外，在两个主要定理中，小型集合$ m $都无法避免。 本文的后半部分致力于将上述两个主要结果推广到$ x \subsetλ^g $，并采取了可数雅的$ g $的动作。长块的作用是由块的块扮演的，其域是Følner序列的成员，而x $中的$ x \在块中的分解为块（其中大多数是ergodic），在一致的块状系统的帮助下获得。
Consider a subshift over a finite alphabet, $X\subset Λ^{\mathbb Z}$ (or $X\subsetΛ^{\mathbb N_0}$). With each finite block $B\inΛ^k$ appearing in $X$ we associate the empirical measure ascribing to every block $C\inΛ^l$ 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 $μ$ 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 $\varepsilon>0$, 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 $\varepsilon$, all blocks in the decomposition are ergodic. 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), after ignoring a set $M$ of upper Banach density smaller than $\varepsilon$, 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 $X\subsetΛ^G$ 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 is ergodic) is obtained with the help of a congruent system of tilings.
