We study the topological complexities of relative entropy zero extensions acted by countableinfinite amenable groups. Firstly, for a given Folner sequence $\{F_n\}_{n=0}^\infty$, we define respectively the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. Meanwhile, we investigate the relations among them. Secondly, we introduce the notion of a relative dimension set. Moreover, using it, we discuss the disjointness between the relative entropy zero extensions which generalizes the results of Dou, Huang and Park[Trans. Amer. Math. Soc. 363(2) (2011), 659-680].
We investigate an analogue of the L^2-Betti numbers for amenable linear subshifts. The role of the von Neumann dimension shall be played by the topological entropy.
Let $(X,G)$ be a $G$-action topological system, where $G$ is a countable infinite discrete amenable group and $X$ a compact metric space. In this paper we study the upper capacity entropy and packing entropy for systems with weaker version of specification. We prove that the upper capacity always carries full entropy while there is a variational principle for packing entropy of saturated sets.
Bowen introduced a definition of topological entropy of subset inspired by Hausdorff dimension in 1973 \cite{B}. In this paper we consider the Bowen's entropy for amenable group action dynamical systems and show that under the tempered condition, the Bowen entropy of the whole compact space for a given Følner sequence equals to the topological entropy. For the proof of this result, we establish a variational principle related to the Bowen entropy and the Brin-Katok's local entropy formula for dynamical systems with amenable group actions.
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 Følner sequences; the Hausdorff dimension of an amenable subshift (for certain metric associated to some Følner 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).
In previous work, I introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit change $σ$-algebra.
It is proved that positive entropy implies mean Li-Yorke chaos for a G-system, where G is a countable infinite discrete bi-orderable amenable group. Examples are given for the cases of integer lattice groups and groups of integer unipotent upper triangular matrices.
A probability measure preserving action of a discrete amenable group $G$ is said to be dominant if it is isomorphic to a generic extension of itself. Recently, it was shown that for $G = \mathbb{Z}$, an action is dominant if and only if it has positive entropy and that for any $G$, positive entropy implies dominance. In this paper, we show that the converse also holds for any $G$, i.e. that zero entropy implies non-dominance.
Let $(X,ρ,G)$ be a $G-$action topological system, where $G$ is a countable infinite discrete amenable group and $X$ a compact metric space. We prove a variational principle for topological entropy of saturated sets for systems which have specification and uniform separation properties. As an application, we compute the topological entropy of level sets and irregular sets.
Packing topological entropy is a dynamical analogy of the packing dimension, which can be viewed as a counterpart of Bowen topological entropy. In the present paper, we will give a systematically study to the packing topological entropy for a continuous $G$-action dynamical system $(X,G)$, where $X$ is a compact metric space and $G$ is a countable discrete amenable group. We first prove a variational principle for amenable packing topological entropy: for any Borel subset $Z$ of $X$, the packing topological entropy of $Z$ equals the supremum of upper local entropy over all Borel probability me…
Let $G$ be a non-periodic amenable group. We prove that there does not exist a topological action of $G$ for which the set of ergodic invariant measures coincides with the set of all ergodic measure-theoretic $G$-systems of entropy zero. Previously J. Serafin, answering a question by B. Weiss, proved the same for $G = \mathbb{Z}$.
In the paper we throw the first light on studying systematically the local entropy theory for a countable discrete amenable group action. For such an action, we introduce entropy tuples in both topological and measure-theoretic settings and build the variational relation between these two kinds of entropy tuples by establishing a local variational principle for a given finite open cover. Moreover, based the idea of topological entropy pairs, we introduce and study two special classes of such an action: uniformly positive entropy and completely positive entropy. Note that in the building of the…
The main purpose of this article is to provide a common generalization of the notions of a topological and Kolmogorov-Sinai entropy for arbitrary representations of discrete amenable groups on objects of (abstract) categories. This is performed by introducing the notion of a measurement functor from the category of representations of a fixed amenable group $Γ$ on objects of an abstract category C to the category of representations of $Γ$ on distributive lattices with localization. We develop the entropy theory of representations of $Γ$ on these lattices, and then define the entropy of a repres…
Let $G$ be a countable infinite discrete amenable group.It should be noted that a $G$-system $(X,G)$ naturally induces a $G$-system $(\mathcal{M}(X),G)$, where $\mathcal{M}(X)$ denotes the space of Borel probability measures on the compact metric space $X$ endowed with the weak*-topology. A factor map $π\colon (X,G)\to(Y,G)$ between two $G$-systems induces a factor map $\widetildeπ\colon(\mathcal{M}(X),G)\to(\mathcal{M}(Y),G)$. It turns out that $\widetildeπ$ is open if and only if $π$ is open. When $Y$ is fully supported, it is shown that $π$ has relative uniformly positive entropy if and onl…
Let $G$ be a countable discrete amenable group which acts continuously on a compact metric space $X$ and let $μ$ be an ergodic $G-$invariant Borel probability measure on $X$. For a fixed tempered Følner sequence $\{F_n\}$ in $G$ with $\lim\limits_{n\rightarrow+\infty}\frac{|F_n|}{\log n}=\infty$, we prove the following variational principle: $$h^B(G_μ,\{F_n\})=h_μ(X,G),$$ where $G_μ$ is the set of generic points for $μ$ with respect to $\{F_n\}$ and $h^B(G_μ,\{F_n\})$ is the Bowen topological entropy (along $\{F_n\}$) on $G_μ$. This generalizes the classical result of Bowen in 1973.
We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case $S=\mathbb N$. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a B…
Results of Wyner and Ziv and of Ornstein and Weiss show that if one observes the first k outputs of a finite-valued ergodic process, then the waiting time until this block appears again is almost surely asymptotic to $2^{hk}$, where $h$ is the entropy of the process. We examine this phenomenon when the allowed return times are restricted to some subset of times, and generalize the results to processes parameterized by other discrete amenable groups. We also obtain a uniform density version of the waiting time results: For a process on $s$ symbols, within a given realization, the density of the…
We introduce a new isomorphism-invariant notion of entropy for measure preserving actions of arbitrary countable groups on probability spaces, which we call orbital Rokhlin entropy. It employs Danilenko's orbital approach to entropy of a partition, and is motivated by Seward's recent generalization of Rokhlin entropy from amenable to general groups. A key ingredient in our approach is the use of an auxiliary probability-measure-preserving hyperfinite equivalence relation. Under the assumption of ergodicity of the auxiliary equivalence relation, our main result is a Shannon-McMillan-Breiman poi…
We introduce a notion of entropy for automorphisms of discrete groups which admit amenable actions on a compact space. This entropy is dual to classical topological entropy in the sense that if G is discrete and abelian then our notion of entropy agrees with the topological entropy of the induced automorphism on the (compact) dual group of G. We prove a number of basic properties of this dual entropy and give a few calculations. In particular, we are able to give precise calculations for arbitrary automorphisms of crystallographic groups.
We study properties of the Weyl pseudometric associated with an action of a countable amenable group on a compact metric space. We prove that the topological entropy and the number of minimal subsets of the closure of an orbit are both lower semicontinuous with respect to the Weyl pseudometric. Furthermore, the simplex of invariant measures supported on the orbit closure varies continuously. We apply the Weyl pseudometric to Toeplitz sequences for amenable residually finite groups. We introduce the notion of a regular Toeplitz sequence for an arbitrary amenable residually finite group and demo…
In this paper, we mainly elucidate a close relationship between the topological entropy and mean dimension theory for actions of polynomial growth groups. We show that metric mean dimension and mean Hausdorff dimension of subshifts with respect to the lower rank subgroup are equal to its topological entropy multiplied by the growth rate of the subgroup. Meanwhile, we also prove the above result holds for the rate distortion dimension of subshifts with respect to the lower rank subgroup and measure entropy. Furthermore, some relevant examples are indicated.
For Γa countable amenable group consider those actions of Γas measure-preserving transformations of a standard probability space, written as {T_γ}_{γ\in Γ} acting on (X,{\cal F}, μ). We say {T_γ}_{γ\inΓ} has completely positive entropy (or simply cpe for short) if for any finite and nontrivial partition P of X the entropy h(T,P) is not zero. Our goal is to demonstrate what is well known for actions of \Bbb Z and even \Bbb Z^d, that actions of completely positive entropy have very strong mixing properties. Let S_i be a list of finite subsets of Γ. We say the S_i spread if any particular γ\neq i…
The notion of topological entropy is originally defined for a single action. Later it was extended by Kieffer for arbitrary discrete amenable groups. Recently Friedland defined topological entropy for any discrete group actions amenable or not. Lind, Schmidt and Ward extended Yuzvinskii's addition formula from single action to abelian group actions. A result of Ward and Zhang suggests that such extension might be possible in the amenable case. In this paper we prove that the addition formula can not be extended for groups of finite type with nonzero Euler-characteristcs e.g. for free groups or…
In 1990, Lind, Schmidt and Ward gave a formula for the entropy of certain $\mathbb{Z}^n$-dynamical systems attached to Laurent polynomials $P$, in terms of the (logarithmic) Mahler measure of $P$. We extend the expansive case of their result to the noncommutative setting where $\mathbb{Z}^n$ gets replaced by suitable discrete amenable groups. Generalizing the Mahler measure, Fuglede-Kadison determinants from the theory of group von Neumann algebras appear in the entropy formula.
In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant. We provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to infinity. These results extend previous work by Lind and Ward on p-adic entropy.
We prove that a generic p.m.p. action of a countable amenable group $G$ has scaling entropy that can not be dominated by a given rate of growth. As a corollary, we obtain that there does not exist a topological action of $G$ for which the set of ergodic invariant measures coincides with the set of all ergodic p.m.p. $G$-systems of entropy zero. We also prove that a generic action of a residually finite amenable group has scaling entropy that can not be bounded from below by a given sequence. We also show an example of an amenable group that has such lower bound for every free p.m.p. action.
By proving an amenable version of Katok's entropy formula and handling the quasi tiling techniques, we establish large deviations bounds for countable discrete amenable group actions. This generalizes the classical results of Lai-Sang Young.
We review subbadditivity properties of Shannon entropy, in particular, from the Shearer's inequality we derive the "infimum rule" for actions of amenable groups. We briefly discuss applicability of the "infimum formula" to actions of other groups. Then we pass to topological entropy of a cover. We prove Shearer's inequality for disjoint covers and give counterexamples otherwise. We also prove that, for actions of amenable groups, the supremum over all open covers of the "infimum fomula" gives correct value of topological entropy.
We provide a definition of a $\prec$-asymptotic pair in a topological action of a countable group $G$, where $\prec$ is an order on $G$ of type $\mathbb Z$. We then prove that if $G$ is a countable amenable group and $(X,G)$ is a topological $G$-action of positive entropy, then for every multiorder $(\tilde{\mathcal O},ν,G)$ and $ν$-almost every order $\prec\,\in\tilde{\mathcal O}$ there exists a $\prec$-asympotic pair in $X$. This result is a generalization of the Blanchard-Host-Ruette Theorem for classical topological dynamical systems (actions of~$\mathbb Z$). We also prove that for every c…
It was proved by Brudno that entropy and Kolmogorov complexity for dynamical systems are tightly related. We generalize his results to the case of arbitrary computable amenable group actions. Namely, for an ergodic shift-action, the asymptotic Kolmogorov complexity of a typical point is equal to the Kolmogorov-Sinai entropy of the action. For topological shift actions, the asymptotic Komogorov complexity of every point is bounded from above by the topological entropy, and there is a point attaining this bound.
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 gro…
Let $G$ be a finitely generated amenable group. We study the space of shifts on $G$ over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that more generally the shifts of entropy $c$ are generic in the space of shifts with entropy at least $c$. The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that for every entropy value $c \in [0,\log |A|]$ there is a weakly mixing subshift of $A^G$ with entropy $c$. We also show that the set of strongly irreducible s…
We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. On route, we investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups. We also present a proof due to Weiss for the fact that topologically prime actions of sofic groups have non-positive topological sofic entropy.
Metric mean dimension of amenable group actions: localization and non-uniformity
School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；School of Mathematics, Hefei University of Technology（合肥工业大学数学学院）
Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved it does. We provide examples showing that Chung and Li's result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive act…
Sofic entropy theory is a generalization of the classical Kolmogorov-Sinai entropy theory to actions of large class of non-amenable groups called sofic groups. This is a short introduction with a guide to the literature.
We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $Γ$ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $Γ$-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $Γ_n$-fixed points, where $(Γ_n, n\ge1)$ is a decreasing sequence of finite index…
We prove that every dynamical system $X$ with free action of a countable amenable group $G$ by homeomorphisms has a zero-dimensional extension $Y$ which is faithful and principal, i.e. every $G$-invariant measure $μ$ on $X$ has exactly one preimage $ν$ on $Y$ and the conditional entropy of $ν$ with respect to $X$ is zero. This is a version of an earlier result by T. Downarowicz and D. Huczek, which establishes the existence of zero-dimensional principal and faithful extensions for general actions of the group of integers.
We define the notion of entropy for a cross section of an action of continuous amenable group and relate it to the entropy of the ambient action. As a result, we are able to answer a question of J.P. Thouvenot about completely positive entropy actions.
It is well known that if $G$ is a countable amenable group and $G \curvearrowright (Y, ν)$ factors onto $G \curvearrowright (X, μ)$, then the entropy of the first action must be greater than or equal to the entropy of the second action. In particular, if $G \curvearrowright (X, μ)$ has infinite entropy, then the action $G \curvearrowright (Y, ν)$ does not admit any finite generating partition. On the other hand, we prove that if $G$ is a countable non-amenable group then there exists a finite integer $n$ with the following property: for every probability-measure-preserving action $G \curvearro…
In this work we study the entropies of subsystems of shifts of finite type (SFTs) and sofic shifts on countable amenable groups. We prove that for any countable amenable group $G$, if $X$ is a $G$-SFT with positive topological entropy $h(X) > 0$, then the entropies of the SFT subsystems of $X$ are dense in the interval $[0, h(X)]$. In fact, we prove a "relative" version of the same result: if $X$ is a $G$-SFT and $Y \subset X$ is a subshift such that $h(Y) < h(X)$, then the entropies of the SFTs $Z$ for which $Y \subset Z \subset X$ are dense in $[h(Y), h(X)]$. We also establish analogous resu…
We show that if $G$ is a a countable amenable group with the comparison property, and $X$ is a strongly irreducible $G$-shift satisfying certain aperiodicity conditions, then $X$ factors onto the full $G$-shift over $N$ symbols, so long as the logarithm of $N$ is less than the topological entropy of $G$.
The aim of this paper is to study growth properties of group extensions of hyperbolic dynamical systems, where we do not assume that the extension satisfies the symmetry conditions seen, for example, in the work of Stadlbauer on symmetric group extensions and of the authors on geodesic flows. Our main application is to growth rates of periodic orbits for covers of an Anosov flow: we reduce the problem of counting periodic orbits in an amenable cover $X$ to counting in a maximal abelian subcover $X^{\mathrm{ab}}$. In this way, we obtain an equivalence for the Gurevič entropy: $h(X)=h(X^{\mathrm…
The entropy in dynamical systems was introduced by A. Kolmogorov. Initially dedicated to iterations of one finite measure preserving transformation, the notion was gradually generalized so as to encompass amenable group actions and topological actions. L. Bowen (2008) succeeded in breaking the non-amenable frontier by introducing the sofic entropy. This invariant provides the same services as the classical entropy for the measured actions of the sofic groups (a class which contains the residually finite groups). In 2010, D. Kerr et H. Li established a topological version together with a variat…
Dan Rudolph showed that for an amenable group $Γ$, the generic measure-preserving action of $Γ$ on a Lebesgue space has zero entropy. Here this is extended to nonamenable groups. In fact, the proof shows that every action is a factor of a zero entropy action! This uses the strange phenomena that in the presence of nonamenability, entropy can increase under a factor map. The proof uses Seward's recent generalization of Sinai's Factor Theorem, the Gaboriau-Lyons result and my theorem that for every nonabelian free group, all Bernoulli shifts factor onto each other.
In this paper we study chaotic behavior of actions of a countable discrete group acting on a compact metric space by self-homeomorphisms. For actions of a countable discrete group G, we introduce local weak mixing and Li-Yorke chaos; and prove that local weak mixing implies Li-Yorke chaos if G is infinite, and positive topological entropy implies local weak mixing if G is an infinite countable discrete amenable group. Moreover, when considering a shift of finite type for actions of an infinite countable amenable group G, if the action has positive topological entropy then its homoclinic equiva…
