# Dynamics of compact quantum metric spaces

## TL;DR

This paper studies how integer actions on compact quantum metric spaces affect their structure, providing criteria for the resulting crossed products to remain quantum metric spaces and analyzing continuous variations in these structures.

## Contribution

It introduces new conditions under which crossed product algebras of quantum metric spaces are themselves quantum metric spaces and examines continuous families of automorphisms and their impact.

## Key findings

- Crossed product algebras can be quantum metric spaces under certain criteria.
- Continuous families of automorphisms lead to continuous fields of crossed products.
- Applications include actions on Riemannian manifolds and Lip-isometric actions.

## Abstract

We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact quantum metric space in a natural way. We moreover provide a flexible set of assumptions ensuring that a continuous family of *-automorphisms of a compact quantum metric space, yields a field of crossed product algebras which varies continuously in Rieffel's quantum Gromov-Hausdorff distance. Lastly we show how our results apply to continuous families of Lip-isometric actions on compact quantum metric spaces and to families of diffeomorphisms of compact Riemannian manifolds which vary continuously in the Whitney C^1-topology.