Smooth crossed product of minimal unique ergodic diffeomorphism of odd sphere

Abstract

For minimal unique ergodic diffeomorphisms ${\alpha }_n$ of $S^{2n+1}(n>0)$ and ${\alpha }_m$ of $S^{2m+1}(m>0)$, the $C^{\ast }$-crossed product algebra $C(S^{2n+1}){⋊}_{{\alpha }_n}\mathbb{Z}$ is isomorphic to $C(S^{2m+1}){⋊}_{{\alpha }_m}\mathbb{Z}$ even though $n\ne m$. However, by cyclic cohomology, we show that smooth crossed product algebra $C^{\infty }(S^{2n+1}){⋊}_{{\alpha }_n}\mathbb{Z}$ is not isomorphic to $C^{\infty }(S^{2m+1}){⋊}_{{\alpha }_m}\mathbb{Z}$ if $n\ne m$.