For a locally compact sofic group continuously acting on a compact metric space, we first study the relative sofic entropy and prove an additive inequality relating sofic entropy and relative sofic entropy. Moreover, it is shown that the relative variational principle remains valid in this paper. Secondly, the topological pressure for locally compact sofic group actions is investigated and the variational principle for topological pressure in this sofic context is established. As an application, we show a sufficient condition for a signed measure to be a $G$-invariant measure. These contributions generalize the classical results for countable sofic groups on such spaces.
In this paper, we introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context.
Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational prin…
Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective we develop a more general approach to sofic entropy which produces both measure and topological dynamical invariants, and we establish the variational principle in this context. In the case of residually finite groups we use the variational principle to compute the topological entropy of principal algebraic actions whose defining group ring element is invert…
This is the first part in a series in which sofic entropy theory is generalized to class-bijective extensions of sofic groupoids. Here we define topological and measure entropy and prove invariance. We also establish the variational principle, compute the entropy of Bernoulli shift actions and answer a question of Benjy Weiss pertaining to the isomorphism problem for non-free Bernoulli shifts. The proofs are independent of previous literature.
We prove that a monoid is sofic, in the sense recently introduced by Ceccherini-Silberstein and Coornaert, whenever the J-class of the identity is a sofic group, and the quotients of this group by orbit stabilisers in the rest of the monoid are amenable. In particular, this shows that the following are all sofic: cancellative monoids with amenable group of units; monoids with sofic group of units and finitely many non-units; and monoids with amenable Schutzenberger groups and finitely many L-classes in each D-class. This provides a very wide range of sofic monoids, subsuming most known example…
We prove that if two topologically free and entropy regular actions of countable sofic groups on compact metrizable spaces are continuously orbit equivalent, and each group either (i) contains a w-normal amenable subgroup which is neither locally finite nor virtually cyclic, or (ii) is a non-locally-finite product of two infinite groups, then the actions have the same sofic topological entropy. This fact is then used to show that if two free uniquely ergodic and entropy regular probability-measure-preserving actions of such groups are boundedly orbit equivalent then the actions have the same s…
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…
Random operators, spectral measures, and local empirical convergence in sofic groups
Fachbereich 10 Mathematik und Informatik, Universität Münster（明斯特大学）；Institut für Mathematik, Universität Leipzig（莱比锡大学）；Israel Institute of Advanced Studies (IIAS)（以色列高等研究院）
We extend the definition of topological pressure to locally compact Hausdorff spaces, and we demonstrate a "variational principle" comparing the topological and measure theoretic pressures. Given a continuous $\mathbb{Z}_+^N$-action $T$ over a locally compact Hausdorff space $X$ and a continuous function vanishing at infinity $f \in C_0(X)$, we define topological pressure $P(T,f)$ using open covers of a special type we call "admissible covers". With this topological pressure, we demonstrate that \begin{equation*} P(T,f) = \sup_μ P_μ(T,f), \end{equation*} where the supremum is taken over all $T…
We say that two free probability-measure-preserving actions of countable groups are Shannon orbit equivalent if there is an orbit equivalence between them whose associated cocycle partitions have finite Shannon entropy. We show that if the acting groups are sofic and each has a w-normal amenable subgroup which is neither locally finite nor virtually cyclic then Shannon orbit equivalence implies that the actions have the same maximum sofic entropy. This extends a result of Austin beyond the finitely generated amenable setting and has the consequence that two Bernoulli actions of a group with th…
For a given topological dynamical system $(X,T)$ over a compact set $X$ with a metric $d$, the "variational principle" states that \begin{equation*} \sup_μh_μ(T) = h(T) = h_d(T), \end{equation*} where $h_μ(T)$ is the Kolmogorov-Sinai entropy, with the supremum taken over every $T$-invariant probability measure, $h_d(T)$ is the Bowen entropy, and $h(T)$ is the topological entropy as defined by Adler, Konheim and McAndrew. In [9], the concept of topological entropy was adapted for the case where $T$ is a proper map and $X$ is locally compact separable and metrizable, and the variational principl…
Let $G$ be a group. Let $X$ be a connected algebraic group over an algebraically closed field $K$. Denote by $A=X(K)$ the set of $K$-points of $X$. We study a class of endomorphisms of pro-algebraic groups, namely algebraic group cellular automata over $(G,X,K)$. They are cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by a homomorphism of algebraic groups $X^M \to X$ where $M\subset G$ is a finite memory set of $τ$. Our first result is that when $G$ is sofic, such an algebraic group cellular automaton $τ$ is invertible whenever it is injective and $\text{char}(K)=0…
This paper has a flaw in an argument that uses the weak-* convergence of measures. The paper was replaced by "Entropy and Its Variational Principle for Locally Compact Metrizable Systems", by the same authors.
It is widely known that when $X$ is compact Hausdorff, and when $T: X \to X$ and $f: X \to \mathbb{R}$ are continuous, \begin{equation*} P(T,f) = \sup_{\text{$μ$: Radon probability}} \left( h_μ(T) + \int f\, \mathrm{d}μ\right), \end{equation*} where $P(T,f)$ is the "topological pressure" and $h_μ(T)$ is the measure theoretic entropy of $T$ with respect to $μ$. This result is known as "variational principle". We generalize the concept of "topological pressure" for the case where $X$ is a separable locally compact metric space. Our definitions are quite similar to those used in the compact case.…
Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the i-th and (i+1)-th Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index i. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.
The paper deals with the existence and uniqueness of the solution of the backward stochastic variational inequality: \begin{equation} \left\{\begin{array} {l}-dY_{t}+\partial φ(Y_{t})dt \ni F(t,Y_{t},Z_{t})dt-Z_{t}dB_{t},\;0\leq t<T \\ Y_{T}=η, \end{array} \right.\end{equation} where $F$ satisfies a local boundedness condition.
The first aims of this work are to endorse the advent of finitely additive set functions as equilibrium states and the possibility to replace the metric entropy by an upper semi-continuous map associated to a general variational principle. More precisely, using methods from Convex Analysis, we construct for each generalized convex pressure function an upper semi-continuous entropy-like map (which, in the context of continuous transformations acting on a compact metric space and the topological pressure, turns out to be the upper semi-continuous envelope of the Kolmogorov-Sinai metric entropy),…
We embed Thompson's group $V$ in the mapping class group of a mixing subshift of finite type. Question~6.3 in [Boyle-Chuysurichay, 17] asks whether these mapping class groups are sofic. Our result suggests that this question is difficult to solve at present (at least for some mixing SFTs), since a resolution of it in either direction would solve an open problem in geometric group theory. More generally, we define the notion of a ``veelike action'', and prove that whenever a group acts veelike on a language, it embeds in the mapping class group of some subshift. We show that Thompson's $V$ acts…
Based on the Carathéodory -Pesin structure theory[11], we introduce three notions of topological pressure of a proper map and provide some properties of these notions. For the proper map of a locally compact separable metric space, we prove some variational principles and give some applications. These are the extensions of results of Pesin, Takens and Verbiski, etc.
