# American Institute of Mathematical Sciences

March  2021, 41(3): 1347-1357. doi: 10.3934/dcds.2020320

## Gromov-Hausdorff stability for group actions

 1 Department of Mathematics, College of Science, Yanbian University, No. 977, Gongyuan Road, Yanji City 133002, Jilin Province, China 2 Department of Mathematics, Chungnam National University, Daejeon 305-764, Korea 3 Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530 21945-970, Rio de Janeiro, Brazil

Received  February 2020 Revised  July 2020 Published  August 2020

Fund Project: KL and MD were supported by the NRF grant funded by the Korea government (MSIT)(NRF-2018R1A2B3001457). CAM was partially supported by the NRF Brain Pool Grant funded by the Korea government and CNPq from Brazil

We will extend the topological Gromov-Hausdorff stability [2] from homeomorphisms to finitely generated actions. We prove that if an action is expansive and has the shadowing property, then it is topologically GH-stable. From this we derive examples of topologically GH-stable actions of the discrete Heisenberg group on tori. Finally, we prove that the topological GH-stability is an invariant under isometric conjugacy.

Citation: Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320
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##### References:
 [1] Samir Chowdhury, Facundo Mémoli. Explicit geodesics in Gromov-Hausdorff space. Electronic Research Announcements, 2018, 25: 48-59. doi: 10.3934/era.2018.25.006 [2] Nhan-Phu Chung. Gromov-Hausdorff distances for dynamical systems. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6179-6200. doi: 10.3934/dcds.2020275 [3] Alexanger Arbieto, Carlos Arnoldo Morales Rojas. Topological stability from Gromov-Hausdorff viewpoint. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3531-3544. doi: 10.3934/dcds.2017151 [4] Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020 [5] Brandon Seward. Every action of a nonamenable group is the factor of a small action. Journal of Modern Dynamics, 2014, 8 (2) : 251-270. doi: 10.3934/jmd.2014.8.251 [6] Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3725-3757. doi: 10.3934/dcds.2021014 [7] Jaeyoo Choy, Hahng-Yun Chu. On the dynamics of flows on compact metric spaces. Communications on Pure & Applied Analysis, 2010, 9 (1) : 103-108. doi: 10.3934/cpaa.2010.9.103 [8] S. A. Krat. On pairs of metrics invariant under a cocompact action of a group. Electronic Research Announcements, 2001, 7: 79-86. [9] Joseph Auslander, Xiongping Dai. Minimality, distality and equicontinuity for semigroup actions on compact Hausdorff spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4647-4711. doi: 10.3934/dcds.2019190 [10] Feimin Zhong, Jinxing Xie, Jing Jiao. Solutions for bargaining games with incomplete information: General type space and action space. Journal of Industrial & Management Optimization, 2018, 14 (3) : 953-966. doi: 10.3934/jimo.2017084 [11] Xiaojun Huang, Yuan Lian, Changrong Zhu. A Billingsley-type theorem for the pressure of an action of an amenable group. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 959-993. doi: 10.3934/dcds.2019040 [12] Carlos Matheus, Jean-Christophe Yoccoz. The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis. Journal of Modern Dynamics, 2010, 4 (3) : 453-486. doi: 10.3934/jmd.2010.4.453 [13] Evgeny L. Korotyaev. Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 219-225. doi: 10.3934/dcds.2011.30.219 [14] Jordi Gaset, Narciso Román-Roy. New multisymplectic approach to the Metric-Affine (Einstein-Palatini) action for gravity. Journal of Geometric Mechanics, 2019, 11 (3) : 361-396. doi: 10.3934/jgm.2019019 [15] Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431 [16] Michael Field, Ian Melbourne, Matthew Nicol, Andrei Török. Statistical properties of compact group extensions of hyperbolic flows and their time one maps. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 79-96. doi: 10.3934/dcds.2005.12.79 [17] Anton Stolbunov. Constructing public-key cryptographic schemes based on class group action on a set of isogenous elliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 215-235. doi: 10.3934/amc.2010.4.215 [18] Qinian Jin, YanYan Li. Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space. Discrete & Continuous Dynamical Systems, 2006, 15 (2) : 367-377. doi: 10.3934/dcds.2006.15.367 [19] Paolo Perfetti. Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 379-391. doi: 10.3934/dcds.1998.4.379 [20] Mickaël D. Chekroun, Jean Roux. Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 3957-3980. doi: 10.3934/dcds.2013.33.3957

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