Gromov-Hausdorff distances for dynamical systems
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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.
We provide examples to illustrate our main results. We change the statement of Corollary 1.4. We also correct typos
Gromov-Hausdorff distance, amenable group, invariant measure, Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Dynamical Systems (math.DS), stability, Stability theory for smooth dynamical systems, pseudo-orbit tracing property, expansiveness, Wasserstein space, FOS: Mathematics, Mathematics - Dynamical Systems, group action, Smooth ergodic theory, invariant measures for smooth dynamical systems
Gromov-Hausdorff distance, amenable group, invariant measure, Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Dynamical Systems (math.DS), stability, Stability theory for smooth dynamical systems, pseudo-orbit tracing property, expansiveness, Wasserstein space, FOS: Mathematics, Mathematics - Dynamical Systems, group action, Smooth ergodic theory, invariant measures for smooth dynamical systems
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