May  2019, 39(5): 2743-2761. doi: 10.3934/dcds.2019115

Topological stability and shadowing of zero-dimensional dynamical systems

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro, Tokyo 153-8914, Japan

Received  May 2018 Revised  September 2018 Published  January 2019

In this paper, we examine the notion of topological stability and its relation to the shadowing properties in zero-dimensional spaces. Several counter-examples on the topological stability and the shadowing properties are given. Also, we prove that any topologically stable (in a modified sense) homeomorphism of a Cantor space exhibits only simple typical dynamics.

Citation: Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115
References:
[1]

E. Akin, The General Topology of Dynamical Systems, Graduate Studies in Mathematics, 1. American Mathematical Society, Providence, RI, 1993. x+261 pp. Google Scholar

[2]

E. AkinE. Glasner and B. Weiss, Generically there is but one self homeomorphism of the Cantor set, Trans. Amer. Math. Soc., 360 (2008), 3613-3630. doi: 10.1090/S0002-9947-08-04450-4. Google Scholar

[3]

N. Aoki and K. Hiraide, Topological theory of dynamical systems, Recent advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co., Amsterdam, 1994. Google Scholar

[4]

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

[5]

J. AuslanderE. Glasner and B. Weiss, On recurrence in zero dimensional flows, Forum Math., 19 (2007), 107-114. doi: 10.1515/FORUM.2007.004. Google Scholar

[6]

L. Blokh and J. Keesling, A characterization of adding machine maps, Topology Appl., 140 (2004), 151-161. doi: 10.1016/j.topol.2003.07.006. Google Scholar

[7]

H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math., 60 (1967), 241-249. doi: 10.4064/fm-60-3-241-249. Google Scholar

[8]

E. Glasner and B. Weiss, The topological Rohlin property and topological entropy, Amer. J. Math., 123 (2001), 1055-1070. doi: 10.1353/ajm.2001.0039. Google Scholar

[9]

M. Hurley, Consequences of topological stability, J. Differential Equations, 54 (1984), 60-72. doi: 10.1016/0022-0396(84)90142-6. Google Scholar

[10]

N. Kawaguchi, Properties of shadowable points: Chaos and equicontinuity, Bull. Braz. Math. Soc. (N.S.), 48 (2017), 599-622. doi: 10.1007/s00574-017-0033-0. Google Scholar

[11]

A. S. Kechris and C. Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc., (3) 94 (2007), 302–350. doi: 10.1112/plms/pdl007. Google Scholar

[12]

T. Kimura, Homeomorphisms of zero-dimensional spaces, Tsukuba J. Math., 12 (1988), 489-495. doi: 10.21099/tkbjm/1496160845. Google Scholar

[13]

P. Kůrka, Topological and Symbolic Dynamics, Societe Mathematique de France, Paris, 2003. Google Scholar

[14]

K. Lee, Continuous inverse shadowing and hyperbolicity, Bull. Austral. Math. Soc., 67 (2003), 15-26. doi: 10.1017/S0004972700033487. Google Scholar

[15]

K. Lee and C. A. Morales, Topological stability and pseudo-orbit tracing property for expansive measures, J. Differential Equations, 262 (2017), 3467-3487. doi: 10.1016/j.jde.2016.04.029. Google Scholar

[16]

R. MetzgerC. A. Morales and P. Thieullen, Topological stability in set-valued dynamics, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1965-1975. doi: 10.3934/dcdsb.2017115. Google Scholar

[17]

T. K. S. Moothathu, Implications of pseudo-orbit tracing property for continuous maps on compacta, Topology Appl., 158 (2011), 2232-2239. doi: 10.1016/j.topol.2011.07.016. Google Scholar

[18]

Z. Nitecki, On semi-stability for diffeomorphisms, Invent. Math., 14 (1971), 83-122. doi: 10.1007/BF01405359. Google Scholar

[19]

Z. Nitecki and M. Shub, Filtrations, decompositions, and explosions, Amer. J. Math., 97 (1975), 1029-1047. doi: 10.2307/2373686. Google Scholar

[20]

T. Shimomura, The pseudo-orbit tracing property and expansiveness on the Cantor set, Proc. Amer. Math. Soc., 106 (1989), 241-244. doi: 10.1090/S0002-9939-1989-0942637-2. Google Scholar

[21]

M. Shub and S. Smale, S. Beyond hyperbolicity, Ann. of Math., (2) 96 (1972), 587–591. doi: 10.2307/1970826. Google Scholar

[22]

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), 231–244, Lecture Notes in Mathematics, 668. Springer, Berlin, 1978. Google Scholar

[23]

K. Yano, Topologically stable homeomorphisms of the circle, Nagoya Math. J., 79 (1980), 145-149. doi: 10.1017/S0027763000018997. Google Scholar

[24]

S. Yu. Pilyugin, The Space of Dynamical Systems with the $C^0$-topology, Lecture Notes in Mathematics, 1571. Springer, Berlin, 1994. doi: 10.1007/BFb0073519. Google Scholar

[25]

S. Yu. Pilyugin and K. Sakai, Shadowing and Hyperbolicity, Lecture Notes in Mathematics, 2193. Springer, Cham, 2017. doi: 10.1007/978-3-319-65184-2. Google Scholar

show all references

References:
[1]

E. Akin, The General Topology of Dynamical Systems, Graduate Studies in Mathematics, 1. American Mathematical Society, Providence, RI, 1993. x+261 pp. Google Scholar

[2]

E. AkinE. Glasner and B. Weiss, Generically there is but one self homeomorphism of the Cantor set, Trans. Amer. Math. Soc., 360 (2008), 3613-3630. doi: 10.1090/S0002-9947-08-04450-4. Google Scholar

[3]

N. Aoki and K. Hiraide, Topological theory of dynamical systems, Recent advances, North-Holland Mathematical Library, 52. North-Holland Publishing Co., Amsterdam, 1994. Google Scholar

[4]

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

[5]

J. AuslanderE. Glasner and B. Weiss, On recurrence in zero dimensional flows, Forum Math., 19 (2007), 107-114. doi: 10.1515/FORUM.2007.004. Google Scholar

[6]

L. Blokh and J. Keesling, A characterization of adding machine maps, Topology Appl., 140 (2004), 151-161. doi: 10.1016/j.topol.2003.07.006. Google Scholar

[7]

H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math., 60 (1967), 241-249. doi: 10.4064/fm-60-3-241-249. Google Scholar

[8]

E. Glasner and B. Weiss, The topological Rohlin property and topological entropy, Amer. J. Math., 123 (2001), 1055-1070. doi: 10.1353/ajm.2001.0039. Google Scholar

[9]

M. Hurley, Consequences of topological stability, J. Differential Equations, 54 (1984), 60-72. doi: 10.1016/0022-0396(84)90142-6. Google Scholar

[10]

N. Kawaguchi, Properties of shadowable points: Chaos and equicontinuity, Bull. Braz. Math. Soc. (N.S.), 48 (2017), 599-622. doi: 10.1007/s00574-017-0033-0. Google Scholar

[11]

A. S. Kechris and C. Rosendal, Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. Lond. Math. Soc., (3) 94 (2007), 302–350. doi: 10.1112/plms/pdl007. Google Scholar

[12]

T. Kimura, Homeomorphisms of zero-dimensional spaces, Tsukuba J. Math., 12 (1988), 489-495. doi: 10.21099/tkbjm/1496160845. Google Scholar

[13]

P. Kůrka, Topological and Symbolic Dynamics, Societe Mathematique de France, Paris, 2003. Google Scholar

[14]

K. Lee, Continuous inverse shadowing and hyperbolicity, Bull. Austral. Math. Soc., 67 (2003), 15-26. doi: 10.1017/S0004972700033487. Google Scholar

[15]

K. Lee and C. A. Morales, Topological stability and pseudo-orbit tracing property for expansive measures, J. Differential Equations, 262 (2017), 3467-3487. doi: 10.1016/j.jde.2016.04.029. Google Scholar

[16]

R. MetzgerC. A. Morales and P. Thieullen, Topological stability in set-valued dynamics, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1965-1975. doi: 10.3934/dcdsb.2017115. Google Scholar

[17]

T. K. S. Moothathu, Implications of pseudo-orbit tracing property for continuous maps on compacta, Topology Appl., 158 (2011), 2232-2239. doi: 10.1016/j.topol.2011.07.016. Google Scholar

[18]

Z. Nitecki, On semi-stability for diffeomorphisms, Invent. Math., 14 (1971), 83-122. doi: 10.1007/BF01405359. Google Scholar

[19]

Z. Nitecki and M. Shub, Filtrations, decompositions, and explosions, Amer. J. Math., 97 (1975), 1029-1047. doi: 10.2307/2373686. Google Scholar

[20]

T. Shimomura, The pseudo-orbit tracing property and expansiveness on the Cantor set, Proc. Amer. Math. Soc., 106 (1989), 241-244. doi: 10.1090/S0002-9939-1989-0942637-2. Google Scholar

[21]

M. Shub and S. Smale, S. Beyond hyperbolicity, Ann. of Math., (2) 96 (1972), 587–591. doi: 10.2307/1970826. Google Scholar

[22]

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), 231–244, Lecture Notes in Mathematics, 668. Springer, Berlin, 1978. Google Scholar

[23]

K. Yano, Topologically stable homeomorphisms of the circle, Nagoya Math. J., 79 (1980), 145-149. doi: 10.1017/S0027763000018997. Google Scholar

[24]

S. Yu. Pilyugin, The Space of Dynamical Systems with the $C^0$-topology, Lecture Notes in Mathematics, 1571. Springer, Berlin, 1994. doi: 10.1007/BFb0073519. Google Scholar

[25]

S. Yu. Pilyugin and K. Sakai, Shadowing and Hyperbolicity, Lecture Notes in Mathematics, 2193. Springer, Cham, 2017. doi: 10.1007/978-3-319-65184-2. Google Scholar

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