December  2020, 40(12): 6845-6854. doi: 10.3934/dcds.2020130

Rigidity of random group actions

1. 

Department of Mathematics, Chungnam National University, Daejeon 305-764, Republic of Korea

2. 

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

3. 

Department of Mathematics, Sungkyunkwan University, Suwon, 16419, Republic of Korea

Received  April 2019 Revised  August 2019 Published  December 2020 Early access  February 2020

Fund Project: K.L. was supported by the National Research Foundation (NRF) grant funded by the Korea government (MSIT) (NRF-2018R1A2B3001457). C.A.M. by the NRF Brain Pool Grant funded by the Korea government (No.2018H1D3A2001632) and CNPq 303389/2015-0. J.O. by NRF 2019R1A2C1002150

We prove that if a finitely generated random group action is robustly expansive and has the shadowing property, then it is rigid. We apply this result to analyze the rigidity of certain iterated function systems or actions of the discrete Heisenberg group.

Citation: Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6845-6854. doi: 10.3934/dcds.2020130
References:
[1]

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

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

A. Z. Bahabadi, Shadowing and average shadowing properties for iterated function systems, Georgian Math. J., 22 (2015), 179-184. 

[4]

R. Bowen, ω-limit sets for axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.  doi: 10.1016/0022-0396(75)90065-0.

[5]

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.

[6]

A. H. Dooley and G. Zhang, Local entropy theory of a random dynamical system, Mem. Amer. Math. Soc., 233 (2015), vi+106 pp. doi: 10.1090/memo/1099.

[7]

M. Fatehi Nia, Iterated function systems with the shadowing property, J. Adv. Res. Pure Math., 7 (2015), 83-91. 

[8]

V. M. Gundlach and Y. Kifer, Random hyperbolic systems, Stochastic Dynamics (Bremen, 1997), 117–145, Springer, New York, 1999.

[9]

M. Hirsch, Differential Topology, Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976.

[10]

M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. of Symposium in Pure Math., 14 (1970), Amer. Math. Soc., 133–163.

[11]

H. HuE. Shi and Z. J. Wang, Some ergodic and rigidity properties of discrete Heisenberg group actions, Israel J. Math., 228 (2018), 933-972.  doi: 10.1007/s11856-018-1787-9.

[12]

P. E. Kloeden and M. Rasmussen, Random Dynamical Systems, , Mathematical Surveys and Monographs, 176. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[13]

M. Mirzavaziri, Function valued metric spaces, Surv. Math. Appl., 5 (2010), 321-332. 

[14]

J. Moser, On a theorem of Anosov, J. Differential Equations, 5 (1969), 411-440.  doi: 10.1016/0022-0396(69)90083-7.

[15]

G. D. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math., (1968), 53–104.

[16]

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.

[17]

C. Robinson and A. Verjovsky, Stability of Anosov diffeomorphisms, https://sites.math.northwestern.edu/ clark/publications/anosov/stability.pdf.

[18]

M. Sambarino and J. L. Vieitez, Robustly expansive homoclinic classes are generically hyperbolic, Discrete Contin. Dyn. Syst., 24 (2009), 1325-1333.  doi: 10.3934/dcds.2009.24.1325.

[19]

S. Smale, Differential dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.

[20]

W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774.  doi: 10.1090/S0002-9939-1950-0038022-3.

[21]

P. Walters, On the pseudo-orbit tracing property and its relationship to stability, The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), Lecture Notes in Math., 668, Springer, Berlin, 668 (1978), 231–244.

[22]

L. Wen, Differentiable Dynamical Systems. An Introduction to Structural Stability and Hyperbolicity, , Graduate Studies in Mathematics, 173. American Mathematical Society, Providence, RI, 2016.

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. 

[2]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[3]

A. Z. Bahabadi, Shadowing and average shadowing properties for iterated function systems, Georgian Math. J., 22 (2015), 179-184. 

[4]

R. Bowen, ω-limit sets for axiom A diffeomorphisms, J. Differential Equations, 18 (1975), 333-339.  doi: 10.1016/0022-0396(75)90065-0.

[5]

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.

[6]

A. H. Dooley and G. Zhang, Local entropy theory of a random dynamical system, Mem. Amer. Math. Soc., 233 (2015), vi+106 pp. doi: 10.1090/memo/1099.

[7]

M. Fatehi Nia, Iterated function systems with the shadowing property, J. Adv. Res. Pure Math., 7 (2015), 83-91. 

[8]

V. M. Gundlach and Y. Kifer, Random hyperbolic systems, Stochastic Dynamics (Bremen, 1997), 117–145, Springer, New York, 1999.

[9]

M. Hirsch, Differential Topology, Graduate Texts in Mathematics, No. 33. Springer-Verlag, New York-Heidelberg, 1976.

[10]

M. Hirsch and C. Pugh, Stable manifolds and hyperbolic sets, Proc. of Symposium in Pure Math., 14 (1970), Amer. Math. Soc., 133–163.

[11]

H. HuE. Shi and Z. J. Wang, Some ergodic and rigidity properties of discrete Heisenberg group actions, Israel J. Math., 228 (2018), 933-972.  doi: 10.1007/s11856-018-1787-9.

[12]

P. E. Kloeden and M. Rasmussen, Random Dynamical Systems, , Mathematical Surveys and Monographs, 176. American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/176.

[13]

M. Mirzavaziri, Function valued metric spaces, Surv. Math. Appl., 5 (2010), 321-332. 

[14]

J. Moser, On a theorem of Anosov, J. Differential Equations, 5 (1969), 411-440.  doi: 10.1016/0022-0396(69)90083-7.

[15]

G. D. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math., (1968), 53–104.

[16]

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.

[17]

C. Robinson and A. Verjovsky, Stability of Anosov diffeomorphisms, https://sites.math.northwestern.edu/ clark/publications/anosov/stability.pdf.

[18]

M. Sambarino and J. L. Vieitez, Robustly expansive homoclinic classes are generically hyperbolic, Discrete Contin. Dyn. Syst., 24 (2009), 1325-1333.  doi: 10.3934/dcds.2009.24.1325.

[19]

S. Smale, Differential dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.  doi: 10.1090/S0002-9904-1967-11798-1.

[20]

W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc., 1 (1950), 769-774.  doi: 10.1090/S0002-9939-1950-0038022-3.

[21]

P. Walters, On the pseudo-orbit tracing property and its relationship to stability, The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State Univ., Fargo, N.D., 1977), Lecture Notes in Math., 668, Springer, Berlin, 668 (1978), 231–244.

[22]

L. Wen, Differentiable Dynamical Systems. An Introduction to Structural Stability and Hyperbolicity, , Graduate Studies in Mathematics, 173. American Mathematical Society, Providence, RI, 2016.

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