A dynamical classification for crossed products of fiberwise essentially minimal zero-dimensional dynamical systems

TL;DR

This paper establishes a classification of crossed product C*-algebras derived from fiberwise essentially minimal zero-dimensional dynamical systems based on their K-theory and strong orbit equivalence, with special cases including no periodic points.

Contribution

It introduces a dynamical classification framework linking K-theory and orbit equivalence for these systems, extending to isomorphism of the associated C*-algebras.

Findings

Crossed products have isomorphic K-theory iff systems are strong orbit equivalent.

Classification theorem for systems with no periodic points including algebra isomorphism.

Analysis of K-theory and Bratteli diagrams for these dynamical systems.

Abstract

We prove that crossed products of fiberwise essentially minimal zero-dimensional dynamical systems have isomorphic $ K $-theory if and only if the dynamical systems are strong orbit equivalent. Under the additional assumption that the dynamical systems have no periodic points, this gives a classification theorem including isomorphism of the $ C^* $-algebras as well. We additionally explore the $ K $-theory of such crossed products and the Bratteli diagrams associated to the dynamical systems.