On the dynamical asymptotic dimension of a free $\mathbb Z^d$-action on the Cantor set

On diagonal actions of free group on the Cantor set

We study diagonal actions $φ:\mathbb{F}_2\curvearrowright\partial\mathbb{F}_2\times K$ on the Cantor set which are given by $φ_a=\partial_a\timesα,φ_b=\partial_b\timesβ$. Under some restrictions on $α,β$ we compute $K_*(C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2)$. As an application in the case of $α$ is Denjoy homeomorphism of the Cantor set and $β=id$ we will show that $C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2$ is Kirchberg algebra with $K_*(C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2)=(\mathbb{Z}^\infty,\mathbb{Z}^\infty)$. Also we will check that $C^*$-crosse…

唯一遍历Cantor极小系统的基本群

Fundamental group of uniquely ergodic Cantor minimal systems

引入了唯一遍历Cantor极小$G$ -系统$\mathcal{R}_{G, ϕ}$的基本群${\mathcal F}(\mathcal{R}_{G, ϕ})$，其中$G$是可数离散群.计算了几个唯一遍历Cantor极小$G$ -系统的基本群.证明了若$\mathcal{R}_{G, ϕ}$由一个可生成阿贝尔群的自由作用$ϕ$产生，则存在$\mathbb{R}$的一个有单位元的可数子环$R$，使得$\mathcal{F}(\mathcal{R}_{G, ϕ})=R_{+}^\times$ .我们还考虑了唯一遍历Cantor极小$\mathbb{Z}^n$ -系统的基本群与交叉积$C^*$ -代数$C(X)\rtimes_ϕ \mathbb{Z}^n$的基本群之间的关系。

C*-algebras of minimal dynamical systems of the product of a Cantor set and an odd dimensional sphere

Let β: S^n \to S^n, for n = 2k + 1, k \geq 1, be one of the known examples of a non-uniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (S^n, β) there is a Cantor minimal system (X, α) such that the corresponding product system (X x S^n, αx β) is minimal and the resulting crossed product C*-algebra C(X x S^n) \rtimes_{αx β} \mathbb{Z} is tracially approximately an interval algebra (TAI). This entails classification for such C*-algebras. Moreover, the minimal Cantor system (X, α) is such that each tracial state on C(X x S^n) \rtimes_β \m…

Almost finiteness and homology of certain non-free actions

We show that Cantor minimal $\mathbb{Z}\rtimes\mathbb{Z}_2$-systems and essentially free amenable odometers are almost finite. We also compute the homology groups of Cantor minimal $\mathbb{Z}\rtimes\mathbb{Z}_2$-systems and show that the associated transformation groupoids satisfy the HK conjecture if and only if the action is free.

Finite simple labeled graph $C^*$-algebras of Cantor minimal subshifts

It is now well known that a simple graph $C^*$-algebra $C^*(E)$ of a directed graph $E$ is either AF or purely infinite. In this paper, we address the question of whether this is the case for labeled graph $C^*$-algebras recently introduced by Bates and Pask as one of the generalizations of graph $C^*$-algebras, and show that there exists a family of simple unital labeled graph $C^*$-algebras which are neither AF nor purely infinite. Actually these algebras are shown to be isomorphic to crossed products $C(X)\times_T \mathbb Z$ where the dynamical systems $(X,T)$ are Cantor minimal subshifts. …

Bratteli-Vershik models for partial actions of $\mathbb{Z}$

Let $U$ and $V$ be open subsets of the Cantor set with finite disjoint complements, and let $h:U\to V$ be a homeomorphism with dense orbits. Building from the ideas of Herman, Putnam, and Skau, we show that the partial action induced by $h$ can be realized as the Vershik map on a Bratteli diagram, and that any two such diagrams are equivalent.