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.     

Growth-type invariants for ℤ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript> subshifts of finite type and arithmetical classes of real numbers

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 Extending an unpublished example of Tsirelson (A strange two-dimensional symbolic system, 1992), we show that growth complexities of the form exp (n ^α+o(1)) are possible for non-integer α’s. In terminology of de Carvalho (Port. Math. 54(1):19–40, 1997), 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, 2001).     

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.     

New results on measures of maximal entropy

It has recently been demonstrated that there are strongly irreducible subshifts of finite type with more than one measure of maximal entropy. Here we obtain a number of results concerning the uniqueness of the measure of maximal entropy. In addition, we construct for anyd≥2 andk a strongly irreducible subshift of finite type ind dimensions with exactlyk ergodic (extremal) measures of maximal entropy. Ford≥3, we construct a strongly irreducible subshift of finite type ind dimensions with a continuum of ergodic measures of maximal entropy. Research supported in part by AFOSR grant # 91-0215 and NSF grant # DMS-9103738. Research supported by the Swedish National Science Foundation.     

Some new examples of Gibbs measures

We investigate Gibbs measures on general subshifts. In particular we show the uniqueness of Gibbs measures as equilibrium states and we construct such measures on other spaces than mixing subshifts of finite type or sofic systems.     

Actions of Cartan subgroups

Using affine Bruhat-Tits buildings, we associate certain subshifts of finite type to systems arising from the action of a Cartan subgroup of ap-adic semisimple Chevalley group on compact quotient spaces Γ\G. These are used to study the resulting dynamical systems. Sponsored in part by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Federal Republic of Germany). Research at MSRI supported by NSF grant DMS8505550.     

Dimension de Billingsley d'ensembles saturés

In this Note, the Cajar's formula on the Billingsley dimension of saturated sets is extended to the case of subshifts of finite type for g-measures.     

Statistical Properties of Endomorphisms and Compact Group Extensions

The statistical properties of endomorphisms under the assumptionthat the associated Perron–Frobenius operator is quasicompactare considered. In particular, the central limit theorem, weakinvariance principle and law of the iterated logarithm for sufficientlyregular observations are examined. The approach clarifies therole of the usual assumptions of ergodicity, weak mixing, andexactness. Sufficient conditions are given for quasicompactness of thePerron–Frobenius operator to lift to the correspondingequivariant operator on a compact group extension of the base.This leads to statistical limit theorems for equivariant observationson compact group extensions. Examples considered include compact group extensions of piecewiseuniformly expanding maps (for example Lasota–Yorke maps),and subshifts of finite type, as well as systems that are nonuniformlyexpanding or nonuniformly hyperbolic.     

Finite procedures for sofic systems

A sofic system is a symbolic flow defined by a finite semigroup. We exhibit finite procedures, involving only the defining semigroup, for answering cetain questions about a sofic system and for constructing certain subshifts of finite type associated with a sofic system.     

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.     

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.     

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.

     

Chaos for Subshifts of Finite Type

This paper deals with chaos for subshifts of finite type. We show that for any subshift of finite type determined by an irreducible and aperiodic matrix, there is a finitely chaotic set with full Hausdorff dimension. Moreover, for any subshift of finite type determined by a matrix, we point out that the cases including positive topological entropy, distributional chaos, chaos and Devaney chaos are mutually equivalent.     

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.     

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.     

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.

     

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.     

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.     

The derivatives of equilibrium states

We find some estimates for the derivatives of equilibrium states of subshifts of finite type. We prove the differentiability (with respect to the potential) of integrals of certain discontinuous functions for the equilibrium state of a potential.research supported by CNPq, Brazil     

Shifts of finite type and Fibonacci Harps

We make an explicit connection between the Fibonacci Harp (or Fibonacci String) and two well-known dynamical systems: subshifts of finite type and the baker’s map on the unit interval. In particular, we show that the boundary of the Fibonacci Harp is an embedding of a commonly studied shift of finite type in the unit interval. Moreover, every shift of finite type embeds as the boundary of a lattice harp.