When are crossed products by minimal diffeomorphisms isomorphic?

TL;DR

This survey explores the isomorphism problem for C* and smooth crossed products by minimal diffeomorphisms, highlighting known results, counterexamples, and open questions in the field.

Contribution

It provides a comprehensive overview of the current state and open problems related to the isomorphism of crossed products by minimal diffeomorphisms.

Findings

Counterexamples to the Giordano-Putnam-Skau Theorem in C* crossed products

Many open problems in smooth crossed product isomorphisms

Discussion of the failure of classical analogs in the non-commutative setting

Abstract

This is a survey which discusses the isomorphism problem for both C* and smooth crossed products by minimal diffeomorphisms. For C* crossed products, examples demonstrate the failure of the obvious analog of the Giordano-Putnam-Skau Theorem on minimal homeomorphisms of the Cantor set. For smooth crossed products, there are many open problems.