The purpose of this work is to bound sofic topological entropy of Toeplitz systems over residually finite groups and to prove the Krieger Theorem about attaining arbitrary entropy by the Toeplitz systems. To achieve these results, we discuss certain properties of the sofic topological entropy in the context of finitely indexed normal subgroups of the group. It will help us to formulate results almost independently of the natural sofic approximation sequences of residually finite 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…
We prove a number of identities relating the sofic entropy of a certain class of non-expansive algebraic dynamical systems, the sofic entropy of the Wired Spanning Forest and the tree entropy of Cayley graphs of residually finite groups. We also show that homoclinic points and periodic points in harmonic models are dense under general conditions.
We prove a formula for the sofic entropy of expansive principal algebraic actions of residually finite groups, extending recent work of Deninger and Schmidt.
In this paper, we study the descriptive set theoretic complexity of the equivalence relation of conjugacy of Toeplitz subshifts of a residually finite group $G$. On the one hand, we show that if $G = \mathbb{Z}$, then topological conjugacy on Toeplitz subshifts with separated holes is amenable. In contrast, if $G$ is non-amenable, then conjugacy of Toeplitz $G$-subshifts is a non-amenable equivalence relation. The results were motivated by a general question, asked by Gao, Jackson and Seward, about the complexity of conjugacy for minimal, free subshifts of countable groups.
Given a dynamical system $(X,G)$, the centralizer $C(G)$ denotes the group of all homeomorphisms of $X$ which commute with the action of $G$. This group is sometimes called the automorphism group of the dynamical system $(X,G)$. In this note, we generalize the construction of Bulatek and Kwiatkowski (1992) to $\mathbb Z^d$-Toepltiz systems by identifying a class of $\mathbb Z^d$-Toeplitz systems that have trivial centralizers. We show that this class of $\mathbb Z^d$-Toeplitz with trivial centralizers contains systems with positive topological entropy.
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…
In this paper, we show that for any sequence ${\bf a}=(a_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ and any $ε>0$, there exists a Toeplitz sequence ${\bf b}=(b_n)_{n\in \Z}\in \{1,\ldots,k\}^\mathbb{Z}$ such that the entropy $h({\bf b})\leq 2 h({\bf a})$ and $\lim_{N\to\infty}\frac{1}{2N+1}\sum_{n=-N}^N|a_n-b_n|<ε$. As an application of this result, we reduce Sarnak Conjecture to Toeplitz systems, that is, if the Möbius function is disjoint from any Toeplitz sequence with zero entropy, then the Sarnak conjecture holds.
In this article we study automorphisms of Toeplitz subshifts. Such groups are abelian and any finitely generated torsion subgroup is finite and cyclic. When the complexity is non superlinear, we prove that the automorphism group is, modulo a finite cyclic group, generated by a unique root of the shift. In the subquadratic complexity case, we show that the automorphism group modulo the torsion is generated by the roots of the shift map and that the result of the non superlinear case is optimal. Namely, for any $\varepsilon > 0$ we construct examples of minimal Toeplitz subshifts with complexity…
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.
To a Toeplitz flow $(X,T)$ we associate an ordered $K^0$-group, denoted $K^0(X,T)$, which is order isomorphic to the $K^0$-group of the associated (non-commutative) $C^\ast$-crossed product $C(X)\rtimes_T \mathbb{Z}$. However, $K^0(X,T)$ can be defined in purely dynamical terms, and it turns out to be a complete invariant for (strong) orbit equivalence. We characterize the $K^0$-groups that arise from Toeplitz flows $(X,T)$ as exactly those simple dimension groups $(G,G^+)$ that contain a noncyclic subgroup $H$ of rank one that intersects $G^+$ nontrivially. Furthermore, the Bratteli diagram r…
Using periodic points we study a notion of entropy with values in the p-adic numbers. This is done for actions of countable discrete residually finite groups $Γ$. For suitable $Γ= \mathbb{Z}^d$-actions we obtain p-adic analogues of multivariable Mahler measures. For certain actions of more general groups the p-adic entropy can be expressed in terms of a p-adic analogue of the Fuglede-Kadison determinant from the theory of von Neumann algebras. Many basic questions remain open.
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…
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…
We relate Fuglede-Kadison determinants to entropy of algebraic actions of sofic groups in essentially complete generality. This generalizes recent results of Hanfeng Li and Andreas Thom from the amenable case to the sofic case, as well as results of David Kerr, Hanfeng Li, and Lewis Bowen in the residually finite case. Moreover, the proof given is the first calculation of entropy of algebraic actions to avoid a nontrivial determinant approximation. We apply our results to a problem of Christopher Deninger, generalizing results of Hanfeng Li and David Kerr. Finally, we show that in many cases, …
Sofic groups were defined implicitly by Gromov in [Gr99] and explicitly by Weiss in [We00]. All residually finite groups (and hence every linear group) is sofic. The purpose of this paper is to introduce, for every countable sofic group $G$, a family of measure-conjugacy invariants for measure-preserving $G$-actions on probability spaces. These invariants generalize Kolmogorov-Sinai entropy for actions of amenable groups. They are computed exactly for Bernoulli shifts over $G$, leading to a complete classification of Bernoulli systems up to measure-conjugacy for many groups including all count…
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…
In 1987, Ornstein and Weiss discovered that the Bernoulli $2$-shift over the rank two free group factors onto the seemingly larger Bernoulli $4$-shift. With the recent creation of an entropy theory for actions of sofic groups (in particular free groups), their example shows the surprising fact that entropy can increase under factor maps. In order to better understand this phenomenon, we study a natural generalization of the Ornstein--Weiss map for countable groups. We relate the increase in entropy to the cost and to the first $\ell^2$-Betti number of the group. More generally, we study coboun…
We continue our study of when topological and measure-theoretic entropy agree for algebraic action of sofic groups. Specifically, we provide a new abstract method to prove that an algebraic action is strongly sofic. The method is based on passing to a "limiting object" for sequences of measures which are asymptotically supported on topological microstates. This limiting object has a natural poset structure that we are able to exploit to prove a max-min principle: if the sofic approximation has ergodic centralizer, then the largest subgroup on which the action is a local weak$^{*}$-limit of mea…
