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Register a new userSymbolic extensions of amenable group actions and the comparison property
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Symbolic Extension Entropy Theorem (SEET) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift. It gives an estimate on the entropy of symbolic extensions (and the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov--Sinai entropy is the estimate, in the topological setup the task reaches beyond the classical theory of entropy. Tools from an extended theory of entropy structures are needed. The main goal of this paper is to prove the SEET for actions of countable amenable groups: Let a countable amenable group $G$ act by homeomorphisms on a compact metric space $X$ and let $mathcal M_G(X)$ denote the simplex of $G$-invariant probability measures on $X$. A function $E $ on $mathcal M_G(X)$ equals the extension entropy function $h^pi$ of a symbolic extension $pi:(Y,G)to (X,G)$, where $h^pi(mu)=sup{h_ u(Y,G): uinpi^{-1}(mu)}$ ($muinmathcal M_G(X)$), if and only if $E $ is an affine superenvelope of the entropy structure of $(X,G)$. The statement is preceded by presentation of the concepts of an entropy structure and superenvelopes, adapted from $mathbb Z$-actions. In full generality we prove a slightly weaker version of SEET, in which symbolic extensions are replaced by quasi-symbolic extensions, i.e., extensions in form of a joining of a subshift with a zero-entropy tiling system. The notion of a tiling system is a subject of earlier works and in this paper we review and complement the theory developed there. The full version of the SEET is proved for groups which are either residually finite or enjoy the comparison property. In order to describe the range of our theorem, we devote a large portion of the paper to the comparison property. Our main result in this aspect shows that all subexponential groups have the comparison property (and thus satisfy the SEET).
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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.
The paper offers a thorough study of multiorders and their applications to measure-preserving actions of countable amenable groups. By a multiorder on a countable group we mean any probability measure $ u$ on the collection $mathcal O$ of linear orders of type $mathbb Z$ on $G$, invariant under the natural action of $G$ on such orders. Every free measure-preserving $G$-action $(X,mu,G)$ has a multiorder $(mathcal O, u,G)$ as a factor and has the same orbits as the $mathbb Z$-action $(X,mu,S)$, where $S$ is the successor map determined by the multiorder factor. The sub-sigma-algebra $Sigma_{mathcal O}$ associated with the multiorder factor is invariant under $S$, which makes the corresponding $mathbb Z$-action $(mathcal O, u,tilde S)$ a factor of $(X,mu,S)$. We prove that the entropy of any $G$-process generated by a finite partition of $X$, conditional with respect to $Sigma_{mathcal O}$, is preserved by the orbit equivalence with $(X,mu,S)$. Furthermore, this entropy can be computed in terms of the so-called random past, by a formula analogous to the one known for $mathbb Z$-actions. This fact is applied to prove a variant of a result by Rudolph and Weiss. The original theorem states that orbit equivalence between free actions of countable amenable groups preserves conditional entropy with respect to a sub-sigma-algebra $Sigma$, as soon as the orbit change is $Sigma$-measurable. In our variant, we replace the measurability assumption by a simpler one: $Sigma$ should be invariant under both actions and the actions on the resulting factor should be free. In conclusion we prove that the Pinsker sigma-algebra of any $G$-process can be identified (with probability 1) using the following algorithm: (1) fix an arbitrary multiorder on $G$, (2) select any order from the support of that multiorder, (3) in the process, find the remote past along the selected order.
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 short note, for countably infinite amenable group actions, we provide topological proofs for the following results: Bowen topological entropy (dimensional entropy) of the whole space equals the usual topological entropy along tempered F{o}lner sequences; the Hausdorff dimension of an amenable subshift (for certain metric associated to some F{o}lner sequence) equals its topological entropy. This answers questions by Zheng and Chen (Israel Journal of Mathematics 212 (2016), 895-911) and Simpson (Theory Comput. Syst. 56 (2015), 527-543).
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.
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