Smooth crossed product of minimal unique ergodic diffeomorphism of odd sphere

Abstract

For minimal unique ergodic diffeomorphisms $\\alphan$ of $S^{2n+1} (n > 0)$ and $\\alpha_m$ of $S^{2m+1}(m>0)$, the $C^\*$-crossed product algebra $C(S^{2n+1})\\rtimes{\\alphan} \\mathbb{Z}$ is isomorphic to $C(S^{2m+1})\\rtimes{\\alpham} \\mathbb{Z}$ even though $n\\neq m$. However, by cyclic cohomology, we show that smooth crossed product algebra $C^\\infty(S^{2n+1})\\rtimes{\\alphan} \\mathbb{Z}$ is not isomorphic to $C^\\infty(S^{2m+1})\\rtimes{\\alpha_m} \\mathbb{Z}$ if~$n\\neq m$.