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Register a new userDiscrete spectrum for amenable group actions
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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.
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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 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.
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 $([0,1]^d)^G$.
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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