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Register a new userContinua having distal minimal actions by amenable groups
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Enhui Shi
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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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We show that if $G$ is a finitely generated group of subexponential growth and $X$ is a Suslinian continuum, then any action of $G$ on $X$ cannot be expansive.
Let $Gamma$ be a lattice in ${rm SL}(n, mathbb R)$ with $ngeq 3$ and $mathcal S$ be a closed surface. Then $Gamma$ has no distal minimal action on $mathcal S$.
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.
Let $X$ be a Peano continuum having a free arc and let $C^0(X)$ be the semigroup of continuous self-maps of $X$. A subsemigroup $Fsubset C^0(X)$ is said to be sensitive, if there is some constant $c>0$ such that for any nonempty open set $Usubset X$, there is some $fin F$ such that the diameter ${rm diam}(f(U))>c$. We show that if $X$ admits a sensitive commutative subsemigroup $F$ of $C^0(X)$ consisting of continuous open maps, then either $X$ is an arc, or $X$ is a circle.
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