# American Institute of Mathematical Sciences

July  2017, 37(7): 3531-3544. doi: 10.3934/dcds.2017151

## Topological stability from Gromov-Hausdorff viewpoint

 Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530, 21945-970 Rio de Janeiro, Brazil

Received  February 2017 Revised  March 2017 Published  March 2019

Fund Project: Work partially supported by CNPq from Brazil.

We combine the classical Gromov-Hausdorff metric [5] with the $C^0$ distance to obtain the $C^0$-Gromov-Hausdorff distance between maps of possibly different metric spaces. The latter is then combined with Walters's topological stability [11] to obtain the notion of topologically GH-stable homeomorphism. We prove that there are topologically stable homeomorphism which are not topologically GH-stable. Also that every topological GH-stable circle homeomorphism is topologically stable. Afterwards, we prove that every expansive homeomorphism with the pseudo-orbit tracing property of a compact metric space is topologically GH-stable. This is related to Walters's stability theorem [11]. Finally, we extend the topological GH-stability to continuous maps and prove the constant maps on compact homogeneous manifolds are topologically GH-stable.

Citation: Alexanger Arbieto, Carlos Arnoldo Morales Rojas. Topological stability from Gromov-Hausdorff viewpoint. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3531-3544. doi: 10.3934/dcds.2017151
##### References:

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##### References:
Topologically but not topologically GH-stable homeomorphism
Isometric stability in $S^1$ implies topological stability
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