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

Gromov-Hausdorff stability for group actions

Meihua Dong 1, , Keonhee Lee 2, and  Carlos Morales 3,

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  March 2021 Early access  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.

Keywords: Gromov-Hausdorff, compact metric space, group action.
Mathematics Subject Classification: Primary: 37B25; Secondary: 53C23.
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
References:
[1]

D. V. Anosov, Roughness of geodesic flows on compact Riemannian manifolds of negative curvature, Dokl. Akad. Nauk SSSR, 145 (1962), 707-709.   Google Scholar

[2]

A. Arbieto and C. A. Morales Rojas, Topological stability from Gromov-Hausdorff viewpoint, Discrete Contin. Dyn. Syst., 37 (2017), 3531-3544.  doi: 10.3934/dcds.2017151.  Google Scholar

[3]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.  Google Scholar

[4]

N.-P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057.  doi: 10.1090/proc/13654.  Google Scholar

[5]

M. Dong, Group Actions from Measure Theoretical Viewpoint, PhD. thesis, Chungnam National University in Daejeon, 2018. Google Scholar

[6]

K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math., 87 (1987), 517-547.  doi: 10.1007/BF01389241.  Google Scholar

[7]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.   Google Scholar

[8]

A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790.  doi: 10.1080/00207179208934253.  Google Scholar

[9]

A. V. Osipov and S. B. Tikhomirov, Shadowing for actions of some finitely generated groups, Dyn. Syst., 29 (2014), 337-351.  doi: 10.1080/14689367.2014.902037.  Google Scholar

[10]

S. Y. Pilyugin and S. B. Tikhomirov, Shadowing in actions of some Abelian groups, Fund. Math., 179 (2003), 83-96.  doi: 10.4064/fm179-1-7.  Google Scholar

[11]

P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in The Structure of Attractors in Dynamical Systems, Lecture Notes in Math., 668, Springer, Berlin, 1978,231–244. doi: 10.1007/BFb0101795.  Google Scholar

show all references

References:
[1]

D. V. Anosov, Roughness of geodesic flows on compact Riemannian manifolds of negative curvature, Dokl. Akad. Nauk SSSR, 145 (1962), 707-709.   Google Scholar

[2]

A. Arbieto and C. A. Morales Rojas, Topological stability from Gromov-Hausdorff viewpoint, Discrete Contin. Dyn. Syst., 37 (2017), 3531-3544.  doi: 10.3934/dcds.2017151.  Google Scholar

[3]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.  Google Scholar

[4]

N.-P. Chung and K. Lee, Topological stability and pseudo-orbit tracing property of group actions, Proc. Amer. Math. Soc., 146 (2018), 1047-1057.  doi: 10.1090/proc/13654.  Google Scholar

[5]

M. Dong, Group Actions from Measure Theoretical Viewpoint, PhD. thesis, Chungnam National University in Daejeon, 2018. Google Scholar

[6]

K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Invent. Math., 87 (1987), 517-547.  doi: 10.1007/BF01389241.  Google Scholar

[7]

M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53-73.   Google Scholar

[8]

A. M. Lyapunov, The general problem of the stability of motion, Internat. J. Control, 55 (1992), 521-790.  doi: 10.1080/00207179208934253.  Google Scholar

[9]

A. V. Osipov and S. B. Tikhomirov, Shadowing for actions of some finitely generated groups, Dyn. Syst., 29 (2014), 337-351.  doi: 10.1080/14689367.2014.902037.  Google Scholar

[10]

S. Y. Pilyugin and S. B. Tikhomirov, Shadowing in actions of some Abelian groups, Fund. Math., 179 (2003), 83-96.  doi: 10.4064/fm179-1-7.  Google Scholar

[11]

P. Walters, On the pseudo-orbit tracing property and its relationship to stability, in The Structure of Attractors in Dynamical Systems, Lecture Notes in Math., 668, Springer, Berlin, 1978,231–244. doi: 10.1007/BFb0101795.  Google Scholar

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