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Register a new userWeak mean equicontinuity for a countable discrete amenable group action
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The weak mean equicontinuous properties for a countable discrete amenable group $G$ acting continuously on a compact metrizable space $X$ are studied. It is shown that the weak mean equicontinuity of $(X times X,G)$ is equivalent to the mean equicontinuity of $(X,G)$. Moreover, when $(X,G)$ has full measure center or $G$ is abelian, it is shown that $(X,G)$ is weak mean equicontinuous if and only if all points in $X$ are uniquely ergodic points and the map $x to mu_x^G$ is continuous, where $mu_x^G$ is the unique ergodic measure on ${ol{Orb(x)}, G}$.
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In this paper, we study discrete spectrum of invariant measures for countable discrete amenable group actions. We show that an invariant measure has discrete spectrum if and only if it has bounded measure complexity. We also prove that, discrete spectrum can be characterized via measure-theoretic complexity using names of a partition and the Hamming distance, and it turns out to be equivalent to both mean equicontinuity and equicontinuity in the mean.
In this note, it is shown that several results concerning mean equicontinuity proved before for minimal systems are actually held for general topological dynamical systems. Particularly, it turns out that a dynamical system is mean equicontinuous if and only if it is equicontinuous in the mean if and only if it is Banach (or Weyl) mean equicontinuous if and only if its regionally proximal relation is equal to the Banach proximal relation. Meanwhile, a relation is introduced such that the smallest closed invariant equivalence relation containing this relation induces the maximal mean equicontinuous factor for any system.
For every infinite (countable discrete) amenable group $G$ and every positive integer $d$ we construct a minimal $G$-action of mean dimension $d/2$ which cannot be embedded in the full $G$-shift on $([0,1]^d)^G$.
In this paper we generalize Kingmans sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.
Consider a subshift over a finite alphabet, $Xsubset Lambda^{mathbb Z}$ (or $XsubsetLambda^{mathbb N_0}$). With each finite block $BinLambda^k$ appearing in $X$ we associate the empirical measure ascribing to every block $CinLambda^l$ 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 $Binmathcal U$. In this paper we prove the following main result: Given $varepsilon>0$, every $xin X$ 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 $xin X$ 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 $XsubsetLambda^G$ with the action of a countable amenable group $G$. The role of long blocks is played by blocks whose domains are members of a Fo lner sequence while the decomposition of $xin X$ into blocks (of which majority is ergodic) is obtained with the help of a congruent system of tilings.
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