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Register a new userSub-additive ergodic theorems for countable amenable groups
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In this paper we generalize Kingmans sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.
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We define the topological pressure for any sub-additive potentials of the countable discrete amenable group action and any given open cover. A local variational principle for the topological pressure is established.
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.
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 ${ol{Orb(x)}, G}$.
Let a countable amenable group $G$ act on a zd compact metric space $X$. For two clopen subsets $mathsf A$ and $mathsf B$ of $X$ we say that $mathsf A$ is emph{subequivalent} to $mathsf B$ (we write $mathsf Apreccurlyeq mathsf B$), if there exists a finite partition $mathsf A=bigcup_{i=1}^k mathsf A_i$ of $mathsf A$ into clopen sets and there are elements $g_1,g_2,dots,g_k$ in $G$ such that $g_1(mathsf A_1), g_2(mathsf A_2),dots, g_k(mathsf A_k)$ are disjoint subsets of $mathsf B$. We say that the action emph{admits comparison} if for any clopen sets $mathsf A, mathsf B$, the condition, that for every $G$-invariant probability measure $mu$ on $X$ we have the sharp inequality $mu(mathsf A)<mu(mathsf B)$, implies $mathsf Apreccurlyeq mathsf B$. Comparison has many desired consequences for the action, such as the existence of tilings with arbitrarily good F{o}lner properties, which are factors of the action. Also, the theory of symbolic extensions, known for $mathbb z$-actions, extends to actions which admit comparison. We also study a purely group-theoretic notion of comparison: if every action of $G$ on any zero-dimensional compact metric space admits comparison then we say that $G$ has the emph{comparison property}. Classical groups $mathbb z$ and $mathbb z^d$ enjoy the comparison property, but in the general case the problem remains open. In this paper we prove this property for groups whose every finitely generated subgroup has subexponential growth.
Let $X$ be a non-degenerate connected compact metric space. If $X$ admits a distal minimal action by a finitely generated amenable group, then the first vCech cohomology group $ {check H}^1(X)$ with integer coefficients is nontrivial. In particular, if $X$ is homotopically equivalent to a CW complex, then $X$ cannot be simply connected.
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