On phase transitions for subshifts of finite type

It has recently been shown that a strongly irreducible subshift of finite type in two or more dimensions may have more than one measure of maximal entropy. In this paper we obtain some results on when (i.e. for what kinds of subshifts of finite type) this happens, and when it does not. In particular, we show that the parameter of a certain subshift of finite type introduced by Burton and Steif has a critical value, below which we have a unique measure of maximal entropy, and above which we have non-uniqueness.     

Exponential law for random subshifts of finite type

In this paper we study the distribution of hitting times for a class of random dynamical systems. We prove that for invariant measures with super-polynomial decay of correlations hitting times to dynamically defined cylinders satisfy exponential distribution. Similar results are obtained for random expanding maps. We emphasize that what we establish is a quenched exponential law for hitting times.     

Bowen-Franks groups of reducible bimodal subshifts of finite type

We study the Bowen-Franks groups of subshifts of finite type associated with reducible bimodal periodic kneading sequences pairs.     

Multidimensional random subshifts of finite type and their large deviations

Summary I introduce random multidimensional subshifts of finite type which generalize models of spin-glasses and establish the “almost sure” large deviations bounds for Gibbs measures there. The paper is sequel to [EKW] where the corresponding results were obtained for deterministic multidimensional subshifts of finite type. Partially supported by US-Israel BSF     

On C*-algebras from homoclinic equivalences on subshifts of finite type

Let G be an infinite countable group and A be a finite set.IfΣ?A~G is a strongly irreducible subshift of finite type,we endow a locally compact and Hausdorff topology on the homoclinic equivalence relation■onΣand show that the reduced C^*-algebra C_r^*(■)of■is a unital simple approximately finite(AF)-dimensional C^*-algebra.The shift action G of onΣinduces a canonical automorphism action of G on the C^*-algebra C_r^*(■).We give the notion of noncommutative dynamical entropy invariants for amenable group actions on C^*-algebras,and show that,if G is an amenable group,then the noncommutative topological entropy of the canonical automorphism action of G on C_r^*(■)is equal to the topology entropy of the shift action of G onΣ.We also establish the variational principle with respect to the noncommutative measure entropy and the topological entropy for the C^*-dynamical system(C_r^*(■),G).     

Constant-to-one extensions of shifts of finite type

Any transitive shift of finite type has a transitive constant-to-one extension which is not of finite type.

     

Weak equivalence for shifts of finite type

We derive a computable set of necessary and sufficient conditions for the existence of a homomorphism from one shift of finite type to another. Also we consider an equivalence relation on subshifts, called weak equivalence, which was introduced and studied by Beal and Perrin. We classify arbitrary shifts of finite type up to weak equivalence.     

Hausdorff measure of sets of finite type of one-sided symbolic space

Some estimation formulae and a computation formula for the Hausdorff measure of the sets of finite type of the one-sided symbolic space are given. Project partially supported by the Natural Science Foundation of Guangdong Province and the Foundation of Advanced Research of Zhongshan University.     

Augmented group systems and shifts of finite type

Let (G, χ, x) be a triple consisting of a finitely presented groupG, epimorphism χ:G →Z, and distinguished elementx ∈G such that χ(x)=1. Given a finite symmetric groupS _r, we construct a finite directed graph Γ that describes the set Φ _r of representations π: Ker χ →S _r as well as the mapping σ _x :Φ _r →Φ _r defined by (σ _x ϱ)(a) = ϱ(x ⁻¹ ax) for alla ∈ Ker χ. The pair (Φ _r ,σ _x has the structure of a shift of finite type, a well-known type of compact 0-dimensional dynamical system. We discuss basic properties and applications of therepresentation shift (Φ _r ,σ _x ), including applications to knot theory.     

The Rohlin property for shifts of finite type

We show that an automorphism of a unital AF C^*-algebra with a certain approximate Rohlin property has the Rohlin property. This generalizes a result of Kishimoto. Using this we show that the shift automorphism on the bilateral C^*-algebra associated with an aperiodic irreducible shift of finite type has the Rohlin property.     

Central limit asymptotics for shifts of finite type

We study the rate of convergence and asymptotic expansions in the central limit theorem for the class of Hölder continuous functions on a shift of finite type endowed with a stationary equilibrium state. It is shown that the rate of convergence in the theorem isO(n ^?1/2) and when the function defines a non-lattice distribution an asymptotic expansion to the order ofo(n ^?1/2) is given. Higher-order expansions can be obtained for a subclass of functions. We also make a remark on the central limit theorem for (closed) orbital measures.     

Classification of the finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity

An absolute valued algebra is a non-zero real algebra that is equipped with a multiplicative norm. We classify all finite dimensional absolute valued algebras having a non-zero central idempotent or a one-sided unity, up to algebra isomorphism. This completes earlier results of Ramírez Álvarez and Rochdi which, in our self-contained presentation, are recovered from the wider context of composition k-algebras with an LR-bijective idempotent.