Two-sided Subshift Research Articles

An étale equivalence relation on a tiling space arising from a two-sided subshift and associated C*-algebras

A λ-graph bisystem consists of a pair of two labelled Bratteli diagrams, that presents a two-sided subshift . We will construct a compact totally disconnected metric space consisting of tilings of a two-dimensional half plane from a λ-graph bisystem. The tiling space has a certain AF-equivalence relation written with a natural shift homeomorphism coming from the shift homeomorphism on the subshift . The equivalence relation yields an AF-algebra with an automorphism induced by . We will study invariance of the étale equivalence relation , the groupoid and the groupoid -algebras , under topological conjugacy of the presenting two-sided subshifts.

-ALGEBRAS ASSOCIATED WITH TWO-SIDED SUBSHIFTS

AbstractThis paper is a continuation of the paper, Matsumoto [‘Subshifts,$\lambda $-graph bisystems and$C^*$-algebras’,J. Math. Anal. Appl.485 (2020), 123843]. A$\lambda $-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying a certain compatibility condition on their edge labeling. For any two-sided subshift$\Lambda $, there exists a$\lambda $-graph bisystem satisfying a special property called the follower–predecessor compatibility condition. We construct an AF-algebra${\mathcal {F}}_{\mathcal {L}}$with shift automorphism$\rho _{\mathcal {L}}$from a$\lambda $-graph bisystem$({\mathcal {L}}^-,{\mathcal {L}}^+)$, and define a$C^*$-algebra${\mathcal R}_{\mathcal {L}}$by the crossed product. It is a two-sided subshift analogue of asymptotic Ruelle algebras constructed from Smale spaces. If$\lambda $-graph bisystems come from two-sided subshifts, these$C^*$-algebras are proved to be invariant under topological conjugacy of the underlying subshifts. We present a simplicity condition of the$C^*$-algebra${\mathcal R}_{\mathcal {L}}$and the K-theory formulas of the$C^*$-algebras${\mathcal {F}}_{\mathcal {L}}$and${\mathcal R}_{\mathcal {L}}$. The K-group for the AF-algebra${\mathcal {F}}_{\mathcal {L}}$is regarded as a two-sided extension of the dimension group of subshifts.

A groupoid approach to C*-algebras associated with λ-graph systems and continuous orbit equivalence of subshifts

ABSTRACT A λ-graph system is a labelled Bratteli diagram with shift operation. It is a generalized notion of finite labelled graph and presents a subshift. We will study continuous orbit equivalence of one-sided subshifts and topological conjugacy of two-sided subshifts from the view points of groupoids and -algebras constructed from λ-graph systems.

Open maps: Small and large holes with unusual properties

Let $X$ be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in $X$. We show that there exist arbitrarily small finite overlapping union of shifted cylinders which intersect every orbit under the shift map. We also show that for any proper subshift $Y$ of $X$ there exists a finite overlapping unions of shifted cylinders such that its survivor set contains $Y$ (in particular, it can have entropy arbitrarily close to the entropy of $X$). Both results may be seen as somewhat counter-intuitive. Finally, we apply these results to a certain class of hyperbolic algebraic automorphisms of a torus.

Diagonal-preserving gauge-invariant isomorphisms of graph C⁎-algebras

We study graph C⁎-algebras equipped with generalised gauge actions, and characterise in terms of groupoids and groupoid cocycles when two graph C⁎-algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the generalised gauge actions. We apply this characterisation to show that two Cuntz–Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding one-sided subshifts are eventually conjugate, and that the stabilisation of two Cuntz–Krieger algebras are isomorphic by a diagonal-preserving isomorphism that intertwines the gauge actions if and only if the corresponding two-sided subshifts are conjugate.

Covering relations and Lyapunov condition for topological conjugacy

In this paper, we study stability of non-autonomous discrete dynamical systems. For a two-sided non-autonomous systems with covering relations determined by a transition matrix A, we show that any small C0 perturbed system has a sequence of compact invariant sets restricted to which the system is topologically semi-conjugate to σA, the two-sided subshift of finite type induced by A. Together with Lyapunov condition of good rate, the semi-conjugacy will become conjugacy. Moreover, if the Lyapunov condition is strict and has perfect rate, then any small C1 perturbed systems is topological conjugate to σA. We also study topological chaos of one-sided systems and systems with limit functions. Lack of hyperbolicity and the time dependence of the rate prevent us from applying classical hyperbolic results or earlier works for autonomous systems: cone condition and Lyapunov function. Two examples are provided to demonstrate the existence of a non-trivial, non-hyperbolic invariant and essential of controlling rate.

Topological dynamics for multidimensional perturbations of maps with covering relations and Liapunov condition

In this paper, we study topological dynamics of high-dimensional systems which are perturbed from a continuous map on R m × R k of the form ( f ( x ) , g ( x , y ) ) . Assume that f has covering relations determined by a transition matrix A. If g is locally trapping, we show that any small C 0 perturbed system has a compact positively invariant set restricted to which the system is topologically semi-conjugate to the one-sided subshift of finite type induced by A. In addition, if the covering relations satisfy a strong Liapunov condition and g is a contraction, we show that any small C 1 perturbed homeomorphism has a compact invariant set restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by A. Some other results about multidimensional perturbations of f are also obtained. The strong Liapunov condition for covering relations is adapted with modification from the cone condition in Zgliczyński (2009) [11]. Our results extend those in Juang et al. (2008) [1], Li et al. (2008) [2], Li and Malkin (2006) [3], Misiurewicz and Zgliczyński (2001) [4] by considering a larger class of maps f and their multidimensional perturbations, and by concluding conjugacy rather than entropy. Our results are applicable to both the logistic and Hénon families.

Subshift attractors of cellular automata

A subshift attractor is a two-sided subshift which is an attractor of a cellular automaton. We prove that each subshift attractor is chain-mixing, contains a configuration which is both F-periodic and σ-periodic and the complement of its language is recursively enumerable. We prove that a subshift of finite type is an attractor of a cellular automaton iff it is mixing. We identify a class of chain-mixing sofic subshifts which are not subshift attractors. We construct a cellular automaton whose maximal attractor is a non-sofic mixing subshift, answering a question raised in Maass (1995 Ergod. Theory Dyn. Syst. 15 663–84). We show that a cellular automaton is surjective on its small quasi-attractor which is the non-empty intersection of all subshift attractors of all Fqσp, where q > 0 and .

Dimension of measures for suspension flows

We consider hyperbolic flows and more generally suspension flows over two-sided subshifts of finite type, and establish an explicit formula for the pointwise dimension of an arbitrary measure (in particular the measure is not necessarily ergodic and it may not possess a local product structure). The applications of this formula include the description of the Hausdorff dimension of a given measure in terms of an ergodic decomposition, and the proof of the existence of measures of maximal dimension.

Stabilized $C^*$-algebras constructed from symbolic dynamical systems

We construct stabilized $C^*$-algebras from subshifts by using the dynamical property of the symbolic dynamical systems. We prove that the construction is dynamical and the $C^*$-algebras are isomorphic to the tensor product $C^*$-algebras between the algebra of all compact operators on a separable Hilbert space and the $C^*$-algebras constructed from creation operators on sub-Fock spaces associated with the subshifts. We also prove that the gauge actions on the stabilized $C^*$-algebras are invariant for topological conjugacy as two-sided subshifts under some conditions. Hence, if two subshifts are topologically conjugate as two-sided subshifts, the associated stabilized $C^*$-algebras are isomorphic so that their K-groups are isomorphic.