One-sided Topological Markov Shifts Research Articles

Finitely presented isomorphisms of Cuntz-Krieger algebras and continuous orbit equivalence of one-sided topological Markov shifts

We introduce the notion of finitely presented isomorphism between Cuntz–Krieger algebras, and of finitely presented isomorphic Cuntz–Krieger algebras. We prove that there exists a finitely presented isomorphism between Cuntz–Krieger algebras $\mathcal{O}_A$ and $\mathcal{O}_B$ if and only if their one-sided topological Markov shifts $(X_A,\sigma_A)$ and $(X_B,\sigma_B)$ are continuously orbit equivalent. Hence the value $\det (I-A)$ is a complete invariant for the existence of a finitely presented isomorphism between isomorphic Cuntz–Krieger algebras, so that there exists a pair of Cuntz–Krieger algebras which are isomorphic but not finitely presented isomorphic.

One-sided topological conjugacy of topological Markov shifts, continuous full groups and Cuntz–Krieger algebras

ABSTRACT We will characterize topologically conjugate one-sided topological Markov shifts in terms of their subgroups of continuous full groups and subalgebras of Cuntz–Krieger algebras.

Subgroups of continuous full groups and relative continuous orbit equivalences of one-sided topological Markov shifts

Let A be an irreducible non-permutation matrix with entries in {0,1}. We study a family ΓA,f of subgroups of the continuous full group ΓA of the one-sided topological Markov shift (XA,σA). The subgroup ΓA,f is indexed by an integer valued continuous function f on XA such that ΓA,0=ΓA the non-amenable continuous full group and ΓA,1=ΓAAF the amenable AF full group. We will first prove that a spatial realization theorem for the subgroups ΓA,f under certain conditions on the functions f. We will second introduce relative versions of continuous orbit equivalence to study classification of the subgroups ΓA,f related to their associated groupoids and C⁎-algebras.

On a family of C*-subalgebras of Cuntz–Krieger algebras

In this paper, we study a family of C*-subalgebras defined by fixed points of generalized gauge actions of a Cuntz–Krieger algebra by introducing a family of étale groupoids whose associated C*-algebras are these C*-subalgebras. We show that topological conjugacy classes of one-sided topological Markov shifts are characterized in terms of the isomorphism classes of these étale groupoids.

On certain inverse semigroups associated with one-sided topological Markov shifts

On certain inverse semigroups associated with one-sided topological Markov shifts

Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts

<p style='text-indent:20px;'>We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.</p>

On one-sided topological conjugacy of topological Markov shifts and gauge actions on Cuntz–Krieger algebras

AbstractWe characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz–Krieger algebras and their gauge actions with potentials.

Coded equivalence of one-sided topological Markov shifts

We introduce a notion of coded equivalence in one-sided topological Markov shifts. The notion is inspired by coding theory. One-sided topological conjugacy implies coded equivalence. We will show that coded equivalence implies continuous orbit equivalence of one-sided topological Markov shifts.

Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz–Krieger algebras

We study KMS states for gauge actions with potential functions on Cuntz--Krieger algebras whose underlying one-sided topological Markov shifts are continuous orbit equivalent. As a result, we have a certain relationship between topological entropy of continuous orbit equivalent one-sided topological Markov shifts.

Compactness of transfer operators and spectral representation of Ruelle zeta functions for super-continuous functions

Transfer operators and Ruelle zeta functions for super-continuous functions on one-sided topological Markov shifts are considered. For every super-continuous function, we construct a Banach space on which the associated transfer operator is compact. Using this Banach space, we establish the trace formula and spectral representation of Ruelle zeta functions for a certain class of super-continuous functions. Our results include, as a special case, the classical trace formula and spectral representation for the class of locally constant functions.

Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz–Krieger algebras

We introduce a notion of uniformly continuous orbit equivalence as a subequivalence relation of continuous orbit equivalence of one-sided topological Markov shifts. It is described in terms of gauge actions on the associated Cuntz–Krieger algebras and continuous full groups of the Markov shifts.

Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras

We will study torus actions on Cuntz--Krieger algebras trivially acting on its canonical maximal abelian $C^*$-subalgebras from the view points of continuous orbit equivalence of one-sided topological Markov shifts and flow equivalence of two-sided topological Markov shifts.

On flow equivalence of one-sided topological Markov shifts

We introduce notions of suspension and flow equivalence on one-sided topological Markov shifts, which we call one-sided suspension and one-sided flow equivalence, respectively. We prove that one-sided flow equivalence is equivalent to continuous orbit equivalence on one-sided topological Markov shifts. We also show that the zeta function of the flow on a one-sided suspension is a dynamical zeta function with some potential function and that the set of certain dynamical zeta functions is invariant under one-sided flow equivalence of topological Markov shifts.

Strongly continuous orbit equivalence of one-sided topological Markov shifts

We will introduce a notion of strongly continuous orbit equivalence in one-sided topological Markov shifts. Strongly continuous orbit equivalence yields a topological conjugacy between their two-sided topological Markov shifts $(\bar{X}_A, \bar{\sigma}_A)$ and $(\bar{X}_B, \bar{\sigma}_B)$. We prove that one-sided topological Markov shifts $(X_A, \sigma_A)$ and $(X_B, \sigma_B)$ are strongly continuous orbit equivalent if and only if there exists an isomorphism bewteen the Cuntz-Krieger algebras ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$ preserving their maximal commutative $C^*$-subalgebras $C(X_A)$ and $C(X_B)$ and giving cocycle conjugate gauge actions. An example of one-sided topological Markov shifts which are strongly continuous orbit equivalent but not one-sided topologically conjugate is presented.

Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions

For continuously orbit equivalent one-sided topological Markov shifts $(X_{A},{\it\sigma}_{A})$ and $(X_{B},{\it\sigma}_{B})$, their eventually periodic points and cocycle functions are studied. As a result, we directly construct an isomorphism between their ordered cohomology groups $(\bar{H}^{A},\bar{H}_{+}^{A})$ and $(\bar{H}^{B},\bar{H}_{+}^{B})$. We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of their ordered cohomology groups, so that the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invariant under continuous orbit equivalence of one-sided topological Markov shifts.

Full groups of one-sided topological Markov shifts

Let $(X_A,\sigma_A)$ be the right one-sided topological Markov shift for an irreducible matrix with entries in $\{0,1\}$, and $\Gamma_A$ the continuous full group of $(X_A,\sigma_A)$. For two irreducible matrices $A$ and $B$ with entries in $\{0,1\}$, it will be proved that the continuous full groups $\Gamma_A$ and $\Gamma_B$ are isomorphic as abstract groups if and only if their one-sided topological Markov shifts $(X_A,\sigma_A)$ and $(X_B,\sigma_B)$ are continuously orbit equivalent.

Continuous orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras

Let A,B be square irreducible matrices with entries in {0,1}. We will show that if the one-sided topological Markov shifts (X_A,\sigma_A) and (X_B,\sigma_B) are continuously orbit equivalent, then the two-sided topological Markov shifts (\bar X_A,\bar\sigma_A) and (\bar X_B,\bar\sigma_B) are flow equivalent, and hence det(id-A)=det(id-B). As a result, the one-sided topological Markov shifts (X_A,\sigma_A) and (X_B,\sigma_B) are continuously orbit equivalent if and only if the Cuntz-Krieger algebras O_A and O_B are isomorphic and det(id-A)=det(id-B).

Classification of Cuntz–Krieger algebras by orbit equivalence of topological Markov shifts

Let A , B A, B be square irreducible matrices with entries in { 0 , 1 } \{0,1 \} . Assume that the determinants of 1 − A 1-A and 1 − B 1-B have the same sign. We will show that the Cuntz–Krieger algebras O A {\mathcal O}_A and O B {\mathcal O}_B are isomorphic if and only if the right one-sided topological Markov shifts ( X A , σ A ) (X_A,\sigma _A) and ( X B , σ B ) (X_B,\sigma _B) are continuously orbit equivalent.

K-groups of the full group actions on one-sided topological Markov shifts

Let $(X_A,\sigma_A)$ be the one-sided topological Markov shift for an $N\times N$ irreducible matrix $A$ with entries in $\{ 0,1 \}$. We will first show that the continuous full group $\Gamma_A$ of $(X_A,\sigma_A)$ is non amenable as a discrete group and contains all finite groups and free groups. We will second introduce the K-group $K^0(X_A,\Gamma_A)$ for the action of $\Gamma_A$ on $X_A$, and show that the group $K^0(X_A,\Gamma_A)$ with the position of the constant $1$ function is a complete invariant for the topological conjugacy class of the action of $\Gamma_A$ on $X_A$ under some conditions.

Orbit equivalence of topological Markov shifts and Cuntz–Krieger algebras

We prove that one-sided topological Markov shifts (X A , σ A ) and (X B , σ B ) for matrices A and B with entries in {0, 1} are continuously orbit equivalent and there exists an isomorphism between the Cuntz-Krieger algebras O A and O B keeping their commutative C*-subalgebras C(X A ) and C(X B ). The part (and hence the only if part) above is equivalent to the condition that there exists a homeomorphism from X A to X B intertwining their topological full groups. We will also study structure of the automorphisms of O A preserving the commutative C * -algebra C(X A ).