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Kengo Matsumoto

Publications and source records attributed to Kengo Matsumoto.

At least 55 records · Page 3Linked to original sources
arXiv2016

On extensions of subshifts by finite groups

$λ$-graph systems are labeled Bratteli diagram with shift operations. They present subshifts. Their matrix presentations are called symbolic matrix systems. We define skew products of $λ$-graph systems and study extensions of subshifts by finite groups. We prove that two canonical symbolic matrix systems are $G$-strong shift equivalent if and only if their presented subshifts are $G$-conjugate. $G$-equivalent classes of subshifts are classified by the cohomology classes of their associated skewing functions.

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arXiv2015

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.

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arXiv2014

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 $(\bar{X}_A, \barσ_A)$ and $(\bar{X}_B, \barσ_B)$. We prove that one-sided topological Markov shifts $(X_A, σ_A)$ and $(X_B, σ_B)$ 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.

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arXiv2014

Pushdown automata, lambda-graph systems and C*-algebras

A $λ$-graph system is a labeled Bratteli diagram with some additional structure, which presents a subshift and yields a $C^*$-algebra. In this paper, we construct a $λ$-graph system from a pushdown automaton, such that the accepted language by the automaton coincides with the language of admissible words of the presented subshift by the $λ$-graph system. The $λ$-graph systems for pushdown automata accepting the languages of Markov-Dyck shifts and sofic-Dyck shifts are presented.

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arXiv2014

On the Markov-Dyck shifts of vertex type

For a given finite directed graph $G$, there are two types of Markov-Dyck shifts, the Markov-Dyck shift $D_G^V$ of vertex type and the Markov-Dyck shift $D_G^E$ of edge type. It is shown that, if $G$ does not have multi-edges, the former is a finite-to-one factor of the latter, and they have the same topological entropy. An expression for the zeta function of a Markov-Dyck shift of vertex type is given. It is different from that of the Markov-Dyck shift of edge type.

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arXiv2014

Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions

For continuously orbit equivalent one-sided topological Markov shifts $(X_A,σ_A)$ and $(X_B,σ_B)$, 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 the ordered cohomology, so that the 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.

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arXiv2013

C*-algebras associated with Hilbert C*-quad modules of finite type

A Hilbert $C^*$-quad module of finite type has a multi structure of Hilbert $C^*$-bimodules with two finite bases. We will construct a $C^*$-algebra from a Hilbert $C^*$-quad module of finite type and prove its universality subject to certain relations among generators. Some examples of the $C^*$-algebras from Hilbert $C^*$-quad modules of finite type will be presented.

math.OA↗
arXiv2013

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,σ_A) and (X_B,σ_B) are continuously orbit equivalent, then the two-sided topological Markov shifts (\bar X_A,\barσ_A) and (\bar X_B,\barσ_B) are flow equivalent, and hence det(id-A)=det(id-B). As a result, the one-sided topological Markov shifts (X_A,σ_A) and (X_B,σ_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).

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arXiv2013

Topological full groups of $C^*$-algebras arising from $β$-expansions

We will introduce a family $Γ_β, 1 < β\in {\mathbb{R}}$ of infinite non-amenable discrete groups as an interpolation of the Higman-Thompson groups $V_n, 1 < n \in {\mathbb{N}}$ by using the topological full groups of the groupoids defined by $β$-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras. The groups $Γ_β, 1 < β\in {\mathbb{R}}$ are realized as groups of piecewise linear functions on $[0,1]$ if the $β$-expansion of $1$ is finite or ultimately periodic. We also classify the groups $Γ_β, 1 < β\in {\mathbb{R}}$ by the number theoretical property of $β$.

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arXiv2012

Full groups of one-sided topological Markov shifts

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

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arXiv2012

$C^*$-algebras associated with textile dynamical systems

A $C^*$-symbolic dynamical system $({\cal A}, ρ, Σ)$ is a finite family $\{ρ_α\}_{α\inΣ}$ of endomorphisms of a $C^*$-algebra ${\cal A}$ with some conditions. It yields a $C^*$-algebra ${\cal O}_ρ$ from an associated Hilbert $C^*$-bimodule. In this paper, we will extend the notion of $C^*$-symbolic dynamical system to $C^*$-textile dynamical system $({\cal A}, ρ, η, {Σ^ρ}, {Σ^η}, κ)$ which consists of two $C^*$-symbolic dynamical systems $({\cal A}, ρ, {Σ^ρ})$ and $({\cal A}, η, {Σ^η})$ with certain commutation relations $κ$ between their endomorphisms $\{ρ_α\}_{α\in Σ^ρ}$ and $\{η_a \}_{a \in Σ^η}$. $C^*$-textile dynamical systems yield two-dimensional subshifts and $C^*$-algebras ${\cal O}^κ_{ρ,η}$. We will study the structure of the algebras ${\cal O}^κ_{ρ,η}$ and present its K-theory formulae.

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arXiv2012

Cuntz-Krieger algebras associated with Hilbert $C^*$-quad modules of commuting matrices

Let ${\cal O}_{{\cal H}^{A,B}_κ}$ be the $C^*$-algebra associated with the Hilbert $C^*$-quad module arising from commuting matrices $A,B$ with entries in $\{0,1\}$. We will show that if the associated tiling space $X_{A,B}^κ$ is transitive, the $C^*$-algebra ${\cal O}_{{\cal H}^{A,B}_κ}$ is simple and purely infinite. In particulr, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^*$-algebra ${\cal O}_{{\cal H}^{[N],[M]}_κ}$ are computed by using the Euclidean algorithm.

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arXiv2011

C*-algebras associated with Hilbert C*-quad modules of C*-textile dynamical systems

A $C^*$-textile dynamical system $({\cal A}, ρ,η,Σ^ρ,Σ^η, κ)$ connsists of a unital $C^*$-algebra ${\cal A}$, two families of endomorphisms ${ρ_α}_{α\in Σ^ρ}$ and ${η_a}_{a \in Σ^η}$ of ${\cal A}$ and certain commutation relations $κ$ among them. It yields a two-dimensional subshift and multi structure of Hilbert $C^*$-bimodules, which we call a Hilbert $C^*$-quad module. We introduce a $C^*$-algebra from the Hilbert $C^*$-quad module as a two-dimensional analogue of Pimsner's construction of $C^*$-algebras from Hilbert $C^*$-bimodules. We study the $C^*$-algebras defined by the Hilbert $C^*$-quad modules and prove that they have universal properties subject to certain operator relations. We also present its examples arising from commuting matrices.

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arXiv2011

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.

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arXiv2011

$C^*$-algebras associated with lambda-synchronizing subshifts and flow equivalence

A certain synchronizing property for subshifts called $λ$-synchronization yields $λ$-graph systems called the $λ$-synchronizing $λ$-graph systems for the subshifts. The $λ$-synchronizing $λ$-graph system is a left Fischer cover analogue for a $λ$-synchronizing subshift. We will study algebraic structure of the $C^*$-algebra associated with the $λ$-synchronizing $λ$-graph system and prove that the stable isomorphism class of the $C^*$-algebra with its Cartan subalgebra is invariant under flow equivalence of $λ$-synchronizing subshifts.

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arXiv2009

Subshifts and C*-algebras from one-counter codes

We introduce a class of subshifts under the name of "standard one-counter shifts". The standard one-counter shifts are the Markov coded systems of certain Markov codes that belong to the family of one-counter languages. We study topological conjugacy and flow equivalence of standard one-counter shifts. To subshifts there are associated C*-algebras by their $λ$-graph systems. We describe a class of standard one-counter shifts with the property that the C*-algebra associated to them is simple, while the C*-algebra that is associated to their inverse is not. This gives examples of subshifts that are not flow equivalent to their inverse. For a family of highly structured standard one-counter shifts we compute the K-groups.

math.DS↗
arXiv2008

A class of simple $C^*$-algebras arising from certain nonsofic subshifts

We present a class of subshifts $Z_N, N = 1,2,...$ whose associated $C^*$-algebras ${\cal O}_{Z_N}$ are simple, purely infinite and not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The class of the subshifts is the first examples whose associated $C^*$-algebras are not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The subshifts $Z_N$ are coded systems whose languages are context free. We compute the topological entropy for the subshifts and show that KMS-state for gauge action on the associated $C^*$-algebra ${\cal O}_{Z_N}$ exists if and only if the logarithm of the inverse temperature is the topological entropy for the subshift $Z_N$, and the corresponding KMS-state is unique.

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