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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[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.

[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.

[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.

[18]

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

[19]

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

[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.

[21]

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

[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.

[23]

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

[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.

[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.

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.

[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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[13]

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

[14]

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

[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.

[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.

[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.

[18]

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

[19]

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

[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.

[21]

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

[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.

[23]

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

[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.

[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.

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