Kengo Matsumoto
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.
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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.
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.
We study discrete flow equivalence of two-sided topological Markov shifts by using extended Ruelle algebras. We characterize flow equivalence of two-sided topological Markov shifts in terms of conjugacy of certain actions weighted by ceiling functions of two-dimensional torus on the stabilized extended Ruelle algebras for the Markov shifts.
Topological entropy in continuous orbit equivalence of one-sided topological Markov shifts
本文证明单侧拓扑马尔可夫移位的连续轨道等价类包含拓扑熵可任意大于或小于给定正实数的移位。
Using Popa's deformation/rigidity theory, we investigate prime decompositions of von Neumann algebras of the form $L(\mathcal{R})$ for countable probability measure preserving equivalence relations $\mathcal{R}$. We show that $L(\mathcal{R})$ is prime whenever $\mathcal{R}$ is nonamenable, ergodic, and admits an unbounded 1-cocycle into a mixing orthogonal representation weakly contained in the regular representation. This is accomplished by constructing the $\textit{Gaussian extension }\tilde{\mathcal{R}}$ of $\mathcal{R}$ and subsequently an s-malleable deformation of the inclusion $L(\mathc…
By the work of Brodzki-Niblo-Nowak-Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence.
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.
We study homeomorphisms of tiling spaces with finite local complexity (FLC), of which suspensions of $d$-dimensional subshifts are an example, and orbit equivalence of tiling spaces with (possibly) infinite local complexity (ILC). In the FLC case, we construct a cohomological invariant of homeomorphisms, and show that all homeomorphisms are a combination of tiling deformations, translations, and local equivalences (MLD). In the ILC case, we construct a cohomological invariant in the so-called weak cohomology, and show that all orbit equivalences are combinations of tiling deformations, transla…
Cohomology for actions of free abelian groups on the Cantor set has (when endowed with an order structure) provided a complete invariance for orbit equivalence. In this paper, we study a particular class of actions of such groups called odometers (or profinite actions) and investigate their cohomology. We show that for a free, minimal $\Z^{d}$-odometer, the first cohomology group provides a complete invariant for the action up to conjugacy. This is in contrast with the situation for orbit equivalence where it is the cohomology in dimension $d$ which provides the invariant. We also consider cla…
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.
In this paper, we will introduce notions of relative version of imprimitivity bimodules and relative version of strong Morita equivalence for pairs of $C^*$-algebras $(\mathcal{A}, \mathcal{D})$ such that $\mathcal{D}$ is a $C^*$-subalgebra of $\mathcal{A}$ with certain conditions. We will then prove that two pairs $(\mathcal{A}_1, \mathcal{D}_1)$ and $(\mathcal{A}_2, \mathcal{D}_2)$ are relatively Morita equivalent if and only if their relative stabilizations are isomorphic. In particularly, for two pairs $(\mathcal{O}_A, \mathcal{D}_A)$ and $(\mathcal{O}_B, \mathcal{D}_B)$ of Cuntz--Krieger …
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).
We will 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 topologically orbit equivalent if and only if there exists an isomorphism between the Cuntz-Krieger algebras ${\Cal O}_A$ and ${\Cal O}_B$ keeping their commutative $C^*$-subalgerbas $C(X_A)$ and $C(X_B)$. It is also 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 ${\Cal O}_A$ keeping the commutative $C^*$-algebra $C(X_A)$.
Quantitative polynomial cohomology and applications to $\textrm L^p$-measure equivalence
本文提出量化多项式上同调，证明nilpotent群的贝蒂数在L^p测度等价下不变，并在简单李群中获得新的消失结果。
A $D_{\infty}$-topological Markov chain can be represented by a pair of zero-one square matrices, which is called a flip pair. We introduce the concepts of $D_{\infty}$-strong shift equivalence and $D_{\infty}$-shift equivalence, which are equivalence relations between flip pairs. We investigate the relationships between the existence of a $D_{\infty}$-conjugacy, the existence of a $D_{\infty}$-strong shift equivalence, the existence of a $D_{\infty}$-shift equivalence and the coincidence of the Lind zeta functions.
We build a bridge between geometric group theory and topological dynamical systems by establishing a dictionary between coarse equivalence and continuous orbit equivalence. As an application, we give conceptual explanations for previous results of Shalom and Sauer on coarse invariance of homological and cohomological dimensions and Shalom's property $H_{FD}$. As another application, we show that group homology and cohomology in a class of coefficients, including all induced and co-induced modules, are coarse invariants. We deduce that being of type $FP_n$ (over arbitrary rings) is a coarse inv…
In the late $70$'s Feldman and Moore defined the cohomology associated to a countable equivalence relation with coefficients in an Abelian Polish group. When the equivalence relation is the orbital one, that is it is induced by a measure preserving action of a countable group $Γ$ on a standard Borel probability space $(X,μ)$, it still makes sense to consider the Feldmann-Moore $1$-cohomology with $G$-coefficients, where this time $G$ can be any topological group. The latter cohomology, denoted by $H^1(Γ\curvearrowright X;G)$, is very misterious and hard to compute, except for some exceptional …
We reinvestigate the theory of deformations of tilings using P-equivariant cohomology. In particular we relate the notion of asymptotically negligible shape functions introduced by Clark and Sadun to weakly P-equivariant forms. We then investigate more closely the relation between deformations of patterns and homeomorphism or topological conjugacy of pattern spaces.
A Complete Invariant for Shift Equivalence for Boolean Matrices and Finite Relations
本文通过周期、回复分支上的诱导偏序及上同调类，构造了布尔矩阵（有限关系）移位等价的完全不变量。
We introduce a notion of abelian cohomology in the context of smooth flows. This is an equivalence relation which is weaker than the standard cohomology equivalence relation for flows. We develop Livshits theory for abelian cohomology over transitive Anosov flows. In particular, we prove an abelian Livshits theorem for homologically full Anosov flows. Then we apply this theorem to strengthen marked length spectrum rigidity for negatively curved surfaces. We also present an application to rigidity of contact Anosov flows. Some new results on homologically full Anosov flows are also given.
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