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.     

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.     

Topological generators of abelian Lie groups and hypercyclic finitely generated abelian semigroups of matrices

In this paper we bring together results about the density of subsemigroups of abelian Lie groups, the minimal number of topological generators of abelian Lie groups and a result about actions of algebraic groups. We find the minimal number of generators of a finitely generated abelian semigroup or group of matrices with a dense or a somewhere dense orbit by computing the minimal number of generators of a dense subsemigroup (or subgroup) of the connected component of the identity of its Zariski closure.     

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.     

Bowen–Franks Groups for Subshifts and Ext-Groups for C*-Algebras

We generalize the Bowen–Franks groups for topological Markov shifts to general subshifts as the Ext-groups for the associated C ^*-algebras. The generalized Bowen–Franks groups for subshifts are shown to be invariant under flow equivalence and, hence, invariant under topological conjugacy. They are regarded as the indices of Fredholm operators related to extensions of the associated C ^*-algebras so that they are described in terms of symbolic dynamical systems. In particular, the group for a sofic subshift is determined by the adjacency matrix of its left Krieger cover graph. The Bowen–Franks groups for some non sofic subshifts are calculated, proving that certain subshifts with the same topological entropy are not flow equivalent.     

Minimal presentations for monoids with the ascending chain condition on principal ideals

We show that the natural way to extend several key results concerning minimal presentations for finitely generated commutative cancellative reduced monoids, is to replace the finitely generated condition by the ascending chain condition on principal ideals.     

Topological complexity, fibrations and symmetry

We show how locally smooth actions of compact Lie groups on a manifold X can be used to obtain new upper bounds for the topological complexity TC(X), in the sense of Farber. We also obtain new bounds for the topological complexity of finitely generated torsion-free nilpotent groups.     

Semigroup actions, covering spaces and Schützenberger groups

We associate a 2-complex to the following data: a presentation of a semigroup S and a transitive action of S on a set V by partial transformations. The automorphism group of the action acts properly discontinuously on this 2-complex. A sufficient condition is given for the 2-complex to be simply connected. As a consequence we obtain simple topological proofs of results on presentations of Schützenberger groups. We also give a geometric proof that a finitely generated regular semigroup with finitely many idempotents has polynomial growth if and only if all its maximal subgroups are virtually nilpotent.     

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.     

Hall’s Theorem for Limit Groups

A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two properties. This answers a question of Sela. Received: May 2006 Revision: May 2007 Accepted: May 2007     

Limits of dihedral groups

We characterise limits of dihedral groups in the space of finitely generated marked groups. We also describe the topological closure of dihedral groups in the space of marked groups on a fixed number of generators.     

A Note on a Chaotic but not S-S Chaotic Subshift

In this paper we prove that the minimal chaotic but not S-S chaotic subshifts are also uinquely ergodic and have zero topological entropy.     

Roundness Properties of Groups

Roundness of metric spaces was introduced by Per Enflo as a tool to study uniform structures of linear topological spaces. The present paper investigates geometric and topological properties detected by the roundness of general metric spaces. In particular, we show that geodesic spaces of roundness 2 are contractible, and that a compact Riemannian manifold with roundness >1 must be simply connected. We then focus our investigation on Cayley graphs of finitely generated groups. One of our main results is that every Cayley graph of a free Abelian group on ⩾ 2 generators has roundness =1. We show that if a group has no Cayley graph of roundness =1, then it must be a torsion group with every element of order 2,3,5, or 7 Partially supported by a Canisius College Summer Research Grant     

Hilbert space compression and exactness of discrete groups

We show that the Hilbert space compression of any (unbounded) finite-dimensional CAT(0) cube complex is 1 and deduce that any finitely generated group acting properly, co-compactly on a CAT(0) cube complex is exact, and hence has Yu's Property A. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)-T(4) small cancellation condition and all those word-hyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams.     

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.     

Uniqueness of Polish group topology

We show the limits of Mackey's theorem applied to identity sets to prove that a given group has a unique Polish group topology.Verbal sets in Abelian Polish groups and full verbal sets in the infinite symmetric group are Borel. However this is not true in general.A Polish group with a neighborhood π-base at 1 of sets from the σ-algebra of identity and verbal sets has a unique Polish group topology. It follows that compact, connected, simple Lie groups, as well as finitely generated profinite groups, have a unique Polish group topology.     

On Bellissima’s construction of the finitely generated free Heyting algebras, and beyond

We study finitely generated free Heyting algebras from a topological and from a model theoretic point of view. We review Bellissima’s representation of the finitely generated free Heyting algebra; we prove that it yields an embedding in the profinite completion, which is also the completion with respect to a naturally defined metric. We give an algebraic interpretation of the Kripke model used by Bellissima as the principal ideal spectrum and show it to be first order interpretable in the Heyting algebra, from which several model theoretic and algebraic properties are derived. In particular, we prove that a free finitely generated Heyting algebra has only one set of free generators, which is definable in it. As a consequence its automorphism group is the permutation group over its generators.     

Development of concept of topological entropy for systems with multiple time

We study the topological entropy for dynamical systems with discrete or continuous multiple time. Due to the generalization of a well-known one time-dimensional result we show that the definition of topological entropy, using the approach for subshifts, leads to the zero entropy for many systems different from subshift. We define a new type of relative topological entropy to avoid this phenomenon. The generalization of Bowen’s power rule allows us to define topological and relative topological entropies for systems with continuous multiple time. As an application, we find a relation between the relative topological entropy and controllability of linear systems with continuous multiple time.