Symmetric Gibbs measures

We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures-a version of de Finetti's theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

     

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     

Mesures de Gibbs invariantes et mesures d'equilibre

We are interested here in the characterization on a symbolic space, of invariant Gibbs measures as equilibrium measures. The first result in this topic was obtained by Lanford and Ruelle (see for example [6]).This problem involves different objects that can all be defined by using the only amenability of the translation group and the only continuity of the local specification. We therefore tried to state our theorems in this general frame. Among the elements of our proof, there is the use of the information gain introduced by H. Föllmer [1] and some arguments similar to those of C.J. Preston in [5]. But the amenability techniques that we widely develop in [2], [4] and [7] are decisive tools for getting the result.The corresponding problem for subshifts is not considered in the present paper so symbolic spaces are product spaces.     

The complexity of the topological conjugacy problem for Toeplitz subshifts

In this paper, we analyze the Borel complexity of the topological conjugacy relation on Toeplitz subshifts. More specifically, we prove that topological conjugacy of Toeplitz subshifts with separated holes is hyperfinite. Indeed, we show that the topological conjugacy relation is hyperfinite on a larger class of Toeplitz subshifts which we call Toeplitz subshifts with growing blocks. This result provides a partial answer to a question asked by Sabok and Tsankov.     

Uniform Szegő cocycles over strictly ergodic subshifts

We consider ergodic families of Verblunsky coefficients generated by minimal aperiodic subshifts. Simon conjectured that the associated probability measures on the unit circle have essential support of zero Lebesgue measure. We prove this statement for a large class of subshifts, namely those satisfying a condition originally introduced by Boshernitzan. This is accomplished by relating the essential support to uniform convergence properties of the corresponding Szegő cocycles.     

Rates of mixing for potentials of summable variation

It is well known that for subshifts of finite type and equilibrium measures associated to Hölder potentials we have exponential decay of correlations. In this article we derive explicit rates of mixing for equilibrium states associated to more general potentials.

     

S-adic conjecture and Bratteli diagrams

In this Note we apply a substantial improvement of a result of S. Ferenczi on S-adic subshifts to give Bratteli–Vershik representations of these subshifts.     

On the connectedness of ergodic systems

The set of ergodic m.p. transformations of the unit interval and the set of ergodic shift-invariant measures on subshifts of finite type are arcwise connected.     

Translation-invariant and periodic Gibbs measures for the Potts model on a Cayley tree

We study translation-invariant Gibbs measures on a Cayley tree of order k = 3 for the ferromagnetic three-state Potts model. We obtain explicit formulas for translation-invariant Gibbs measures. We also consider periodic Gibbs measures on a Cayley tree of order k for the antiferromagnetic q-state Potts model. Moreover, we improve previously obtained results: we find the exact number of periodic Gibbs measures with the period two on a Cayley tree of order k ≥ 3 that are defined on some invariant sets.     

Topological Entropy of a Class of Subshifts of Finite Type

In this paper, we construct a special class of subshifts of finite type. By studying the spectral radius of the transfer matrix associated with the subshift of finite type, we obtain an estimation of its topological entropy. Interestingly, we find that the topological entropy of this class of subshifts of finite type converges monotonically to log(n + 1) (a constant only depends on the structure of the transfer matrices) as the increasing of the order of the transfer matrices.     

On the prevalence of zero entropy

For a residual subset of the space of discrete stationary stochastic processes (under the weak topology for measures) the entropy is 0. The processes with arbitrarily large entropy are dense. For a residual subset of subshifts of the shift on a finite alphabet, the topological entropy is 0.     

A certain synchronizing property of subshifts and flow equivalence

We will study a certain synchronizing property of subshifts called λ-synchronization. The λ-synchronizing subshifts form a large class of irreducible subshifts containing irreducible sofic shifts. We prove that the λ-synchronization is invariant under flow equivalence of subshifts. The λ-synchronizing K-groups and the λ-synchronizing Bowen-Franks groups are studied and proved to be invariant under flow equivalence of λ-synchronizing subshifts. They are new flow equivalence invariants for λ-synchronizing subshifts.     

Sets of “Non-typical” points have full topological entropy and full Hausdorff dimension

For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.     

Set of periods of a subshift

In this article, subsets of \({\mathbb {N}}\) that can arise as sets of periods of the following subshifts are characterized: (i) subshifts of finite type, (ii) transitive subshifts of finite type, (iii) sofic shifts, (iv) transitive sofic shifts, and (v) arbitrary subshifts.     

Toeplitz minimal flows which are not uniquely ergodic

Summary We study minimal symbolic dynamical systems which are orbit closures of Toeplitz sequences. We construct 0–1 subshifts of this type for which the set of ergodic invariant measures has any given finite cardinality, is countably infinite or has cardinality of the continuum.     

A conjugacy invariant for reducible sofic shifts and its semigroup characterizations

We introduce a notion of magic words and, through them, we present a lattice of sub-synchronizing subshifts which describes the synchronizing parts of a sofic shiftS. We show that topological conjugacy maps subsynchronizing subshifts onto sub-synchronizing subshifts, it preserves their mutual relationship (i.e. the corresponding lattices are isomorphic) and the corresponding covers within the Krieger covers are topologically conjugate. Using the magic words, a full characterization of the syntactic monoid of a shift of finite type is given. We show that a synchronizing deterministic presentation of every sub-synchronizing subshift ofS can be seen within a two-sided ideal of the syntactic monoid ofS.     

Periodic Gibbs measures for the Potts model on the Cayley tree

We study the Potts model on the Cayley tree. We demonstrate that for this model with a zero external field, periodic Gibbs measures on some invariant sets are translation invariant. Furthermore, we find the conditions under which the Potts model with a nonzero external field admits periodic Gibbs measures.