Search

Let \Gamma be a sofic group, \Sigma be a sofic approximation sequence of \Gamma and X be a \Gamma-subshift with non-negative sofic topological entropy with respect to \Sigma. Further assume that X is a shift of finite type, or more generally, that X satisfies the topological Markov property. We show that for any sufficiently regular potential f \colon X \to \mathbb {R}, any translation-invariant Borel probability measure on X which maximizes the measure-theoretic sofic pressure of f with respect to \Sigma is a Gibbs state with respect to f. This extends a classical theorem of Lanford and Ruelle, as well as previous generalizations of Moulin Ollagnier, Pinchon, Tempelman and others, to the case where the group is sofic. As applications of our main result we present a criterion for uniqueness of an equilibrium measure, as well as sufficient conditions for having that the equilibrium states do not depend upon the chosen sofic approximation sequence. We also prove that for any group-shift over a sofic group, the Haar measure is the unique measure of maximal sofic entropy for every sofic approximation sequence, as long as the homoclinic group is dense. On the expository side, we present a short proof of Chung's variational principle for sofic topological pressure. [ABSTRACT FROM AUTHOR]

Given a finite generating set A for a group Γ, we study the map W 7→ WA as a topological dynamical system - a continuous self-map of the compact metrizable space of subsets of Γ. If the set A generates Γ as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when Γ D Zd and A Zd is a finite positively generating set containing the identity, the natural invertible extension of the map W 7→ W C A is always topologically conjugate to the unique "north-south" dynamics on the Cantor set. In contrast to this, we show that various natural "geometric" properties of the finitely generated group (Γ; A) can be recovered from the dynamics of this map, in particular, the growth type and amenability of Γ. When Γ = Zd, we show that the volume of the convex hull of the generating set A is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group Γ, related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case. [ABSTRACT FROM AUTHOR]

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 that they do. 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 action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li's algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group. [ABSTRACT FROM AUTHOR]

Mañé proved in 1979 that if a compact metric space admits an expansive homeomorphism, then it is finite dimensional. We generalize this theorem to multiparameter actions. The generalization involves mean dimension theory, which counts the "averaged dimension" of a dynamical system. We prove that if T : Zk ×X → X is expansive and if R : Zk−1 ×X → X commutes with T, then R has finite mean dimension. When k = 1, this statement reduces to Mañé's theorem. We also study several related issues, especially the connection with entropy theory. [ABSTRACT FROM AUTHOR]

This paper considers the general question of when a topological action of a countable group can be factored into a direct product of non-trivial actions. In the early 1980s, D. Lind considered such questions for $\mathbb{Z}$-shifts of finite type. In particular, we study direct factorizations of subshifts of finite type over $\mathbb{Z}^{d}$ and other groups, and $\mathbb{Z}$-subshifts which are not of finite type. The main results concern direct factors of the multidimensional full $n$-shift, the multidimensional $3$-colored chessboard and the Dyck shift over a prime alphabet. A direct factorization of an expansive $\mathbb{G}$-action must be finite, but an example is provided of a non-expansive $\mathbb{Z}$-action for which there is no finite direct-prime factorization. The question about existence of direct-prime factorization of expansive actions remains open, even for $\mathbb{G}=\mathbb{Z}$. [ABSTRACT FROM PUBLISHER]

For subshifts of finite type (SFTs), any equilibrium measure is Gibbs, as long as $f$ has $d$-summable variation. This is a theorem of Lanford and Ruelle. Conversely, a theorem of Dobrušin states that for strongly irreducible subshifts, shift-invariant Gibbs measures are equilibrium measures. Here we prove a generalization of the Lanford–Ruelle theorem: for all subshifts, any equilibrium measure for a function with $d$-summable variation is ‘topologically Gibbs’. This is a relaxed notion which coincides with the usual notion of a Gibbs measure for SFTs. In the second part of the paper, we study Gibbs and equilibrium measures for some interesting families of subshifts: $\beta $-shifts, Dyck shifts and Kalikow-type shifts (defined below). In all of these cases, a Lanford–Ruelle-type theorem holds. For each of these families, we provide a specific proof of the result. [ABSTRACT FROM PUBLISHER]

We discuss some numerical invariants of multidimensional shifts of finite type (SFTs) which are associated with the growth rates of the number of admissible finite configurations. Extending an unpublished example of Tsirelson (A strange two-dimensional symbolic system, ), we show that growth complexities of the form exp ( n) are possible for non-integer α's. In terminology of de Carvalho (Port. Math. 54(1):19-40, ), such subshifts have entropy dimension α. The class of possible α's are identified in terms of arithmetical classes of real numbers of Weihrauch and Zheng (Math. Log. Q. 47(1):51-65, ). [ABSTRACT FROM AUTHOR]