Two minimal unique ergodic diffeomorphisms on a manifolds and their smooth crossed product algebras

TL;DR

This paper constructs two minimal, uniquely ergodic diffeomorphisms on a specific manifold and compares their smooth crossed product algebras, revealing differences in smooth versus continuous cases.

Contribution

It introduces two explicit minimal uniquely ergodic diffeomorphisms with identical continuous crossed products but distinct smooth crossed product algebras.

Findings

Continuous crossed products are isomorphic for both diffeomorphisms.

Smooth crossed products are not isomorphic, highlighting smooth structure sensitivity.

Abstract

In this article we construct two minimal unique ergodic diffeomorphisms $\alpha$ and $\beta$ on $S^3 \times S^{6} \times S^{8} $. We will show that $C(S^3 \times S^{6} \times S^{8}) \rtimes_\alpha \mathbb{Z} $ and $C(S^3 \times S^{6} \times S^{8})\rtimes_\beta \mathbb{Z} $ are equivalent to each other, while $C^\infty (S^3 \times S^{6} \times S^{8})\rtimes_\alpha \mathbb{Z} $ and $C^\infty(S^3 \times S^{6} \times S^{8} )\rtimes_\beta \mathbb{Z} $ are not.