A short proof of affability for certain Cantor minimal $\Z^2$-systems
We will show that any extension of a product of two Cantor minimal $\Z$-systems is affable in the sense of Giordano, Putnam and Skau.
math.DS↗
arXiv subjects
Hiroki Matui
Publications and source records attributed to Hiroki Matui.
Minimal dynamical systems on the product of the Cantor set and the circle II
Let $X$ be the Cantor set and $ϕ$ be a minimal homeomorphism on $X\times\T$. We show that the crossed product $C^*$-algebra $C^*(X\times\T,ϕ)$ is a simple $A\T$-algebra provided that the associated cocycle takes its values in rotations on $\T$. Given two minimal systems $(X\times\T,ϕ)$ and $(Y\times\T,ψ)$ such that $ϕ$ and $ψ$ arise from cocycles with values in isometric homeomorphisms on $\T$, we show that two systems are approximately $K$-conjugate when they have the same $K$-theoretical information.
math.OA↗
Minimal dynamical systems on the product of the Cantor set and the circle
We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if that system is rigid. In the case that cocycles take values in the rotation group, it is also shown that rigidity implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately $K$-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps $C(X\times \T)$ onto $C(X\times \T)$.
math.OA↗
Approximate conjugacy and full groups of Cantor minimal systems
H. Lin and the author introduced the notion of approximate conjugacy of dynamical systems. In this paper, we will discuss the relationship between approximate conjugacy and full groups of Cantor minimal systems. An analogue of Glasner-Weiss's theorem will be shown. Approximate conjugacy of dynamical systems on the product of the Cantor set and the circle will also be studied.
math.DS↗
Minimal Dynamical Systems and Approximate Conjugacy
Several versions of approximate conjugacy for minimal dynamical systems are introduced. Relation between approximate conjugacy and corresponding crossed product $C^*$-algebras is discussed. For the Cantor minimal systems, a complete description is given for these relations via $K$-theory and $C^*$-algebras. For example, it is shown that two Cantor minimal systems are approximately $τ$-conjugate if and only if they are orbit equivalent and have the same periodic spectrum. It is also shown that two such systems are approximately $K$-conjugate if and only if the corresponding crossed product $C^*$-algebras have the same scaled ordered $K$-theory. Consequently, two Cantor minimal systems are approximately $K$-conjugate if and only if the associated transformation $C^*$-algebras are isomorphic. Incidentally, this approximate $K$-conjugacy coincides with Giordano, Putnam and Skau's strong orbit equivalence for the Cantor minimal systems.
math.OA↗