Minimal dynamical systems on connected odd dimensional spaces

Abstract

Let $\beta: S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that the crossed product $C(S^{2n+1})\rtimes_{\beta} \Z$ has rational tracial rank at most one. More generally, let $\Omega$ be a connected compact metric space with finite covering dimension and with $H^1(\Omega, \Z)=\{0\}.$ Suppose that $K_i(C(\Omega))=\Z\oplus G_i$ for some finite abelian group $G_i,$ $i=0,1.$ Let $\beta: \Omega\to\Omega$ be a minimal homeomorphism. We also show that $A=C(\Omega)\rtimes_{\beta}\Z$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This was done by studying the minimal homeomorphisms on $X\times \Omega,$ where $X$ is the Cantor set.