# Gromov-Hausdorff distances for dynamical systems

## TL;DR

This paper extends the concept of Gromov-Hausdorff distances to dynamical systems with continuous actions, establishing stability results for expansive actions with pseudo-orbit tracing, and explores convergence in Wasserstein spaces using optimal transport.

## Contribution

It introduces an adapted equivariant Gromov-Hausdorff distance for continuous actions and proves stability under this topology for expansive systems with pseudo-orbit tracing.

## Key findings

- Expansive actions with pseudo-orbit tracing are stable under the new topology.
- Established convergence criteria for group actions on Wasserstein spaces.
- Connected curvature-dimension conditions with equivariant Gromov-Hausdorff convergence.

## Abstract

We study equivariant Gromov-Hausdorff distances for general continuous actions which are not necessarily isometric as Fukaya introduced. We prove that if an action is expansive and has pseudo-orbit tracing property then it is stable under our adapted equivariant Gromov-Hausdorff topology. Finally, using Lott and Villani's ideas of optimal transport in studying curvature-dimension conditions, we investigate equivariant Gromov-Hausdorff convergence for actions of locally compact amenable groups on Wasserstein spaces.