Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-17T11:22:13.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperspace Dynamics of Generic Maps of the Cantor Space

Published online by Cambridge University Press:  20 November 2018

Nilson C. Bernardes Jr.  and
Rômulo M. Vermersch
Nilson C. Bernardes Jr.
Affiliation:
Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21945-970, Brasil. e-mail: bernardes@im.ufrj.br
Rômulo M. Vermersch
Affiliation:
Departamento de Tecnologias e Linguagens, Instituto Multidisciplinar, Universidade Federal Rural do Rio de Janeiro, Nova Iguaçu, RJ, 26020-740, Brasil. e-mail: romulo.vermersch@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study the hyperspace dynamics induced fromgeneric continuous maps and fromgeneric homeomorphisms of the Cantor space, with emphasis on the notions of Li– Yorke chaos, distributional chaos, topological entropy, chain continuity, shadowing, and recurrence.

Keywords

37B9954H2054E52cantor spacecontinuous mapshomeomorphismshyperspacedynamics
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 2 , April 2015 , pp. 330 - 349
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Abdenur, F. and Andersson, M., Ergodic theory of generic continuous maps. Commun. Math. Phys. 318(2013), no. 3, 831– 855.http://dx.doi.org/10.1007/s00220-012-1622-9 Google Scholar
[2] Acosta, G., Illanes, A., and Méndez– Lango, H., The transitivity of induced maps. Topology Appl. 156(2009), no. 5, 1013– 1033. http://dx.doi.org/10.1016/j.topol.2008.12.025 Google Scholar
[3] Adler, R. L., Konheim, A. G., and McAndrew, M. H., Topological entropy. Trans. Amer. Math. Soc. 114(1965), 309– 319. http://dx.doi.org/10.1090/S0002-9947-1965-0175106-9 Google Scholar
[4] Agronsky, S. J., Bruckner, A. M., and Laczkovich, M., Dynamics of typical continuous functions. J. London Math. Soc. (2) 40(1989), no. 2, 227– 243.http://dx.doi.org/10.1112/jlms/s2-40.2.227 Google Scholar
[5] Akin, E., On chain continuity. Discrete Contin. Dynam. Systems 2(1996), no. 1, 111– 120.http://dx.doi.org/10.3934/dcds.1996.2.111 Google Scholar
[6] Akin, E., Glasner, E., and Weiss, B., Generically there is but one self homeomorphism of the Cantor set. Trans. Amer. Math. Soc. 360(2008), no. 7, 3613– 3630.http://dx.doi.org/10.1090/S0002-9947-08-04450-4 Google Scholar
[7] Akin, E., Hurley, M., and Kennedy, J. A., Dynamics of topologically generic homeomorphisms. Mem. Amer. Math. Soc. 164(2003), no. 783.Google Scholar
[8] Bauer, W. and Sigmund, K., Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math. 79(1975), 81– 92.http://dx.doi.org/10.1007/BF01585664 Google Scholar
[9] Bernardes, N. C. Jr., On the predictability of discrete dynamical systems. Proc. Amer. Math. Soc. 130(2002), no. 7, 1983– 1992. http://dx.doi.org/10.1090/S0002-9939-01-06247-5 Google Scholar
[10] Bernardes, N. C. Jr., On the predictability of discrete dynamical systems. II. Proc. Amer. Math. Soc. 133(2005), no. 12, 3473– 3483.http://dx.doi.org/10.1090/S0002-9939-05-07924-4 Google Scholar
[11] Bernardes, N. C. Jr., Limit sets of typical continuous functions. J. Math. Anal. Appl. 319(2006), no. 2, 651– 659. http://dx.doi.org/10.1016/j.jmaa.2005.06.056 Google Scholar
[12] Bernardes, N. C. Jr., On the predictability of discrete dynamical systems. III. J. Math. Anal. Appl. 339(2008), no. 1, 58– 69. http://dx.doi.org/10.1016/j.jmaa.2007.06.029 Google Scholar
[13] Bernardes, N. C. Jr., Limit sets of typical homeomorphisms. Canad. Math. Bull. 55(2012), no. 2, 225– 232. http://dx.doi.org/10.4153/CMB-2011-066-2 Google Scholar
[14] Bernardes, N. C. Jr., Addendum to “Limit sets of typical homeomorphisms”. Canad. Math. Bull., to appear.http://dx.doi.org/10.4153/CMB-2012-033-1 Google Scholar
[15] Bernardes, N. C. Jr.. and Darji, U. B., Graph theoretic structure of maps of the Cantor space. Adv. Math. 231(2012), no. 3– 4, 1655– 1680. http://dx.doi.org/10.1016/j.aim.2012.05.024 Google Scholar
[16] Blanchard, F., Glasner, E., Kolyada, S., and Maass, A., On Li– Yorke pairs. J. Reine Angew. Math. 547(2002), 51– 68.Google Scholar
[17] Bowen, R., Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153(1971), 401– 414. http://dx.doi.org/10.1090/S0002-9947-1971-0274707-X Google Scholar
[18] Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin-New York, 1975.Google Scholar
[19] Bowen, R., ω-limit sets for axiom A diffeomorphisms. J. Differential Equations 18(1975), no. 2, 333– 339.http://dx.doi.org/10.1016/0022-0396(75)90065-0 Google Scholar
[20] Brouwer, L. E. J., On the structure of perfect sets of points. Proc. Akad. Amsterdam 12(1910), 758– 794.Google Scholar
[21] D’Aniello, E. and Darji, U. B., Chaos among self– maps of the Cantor space. J. Math. Anal. Appl. 381(2011), no. 2, 781– 788. http://dx.doi.org/10.1016/j.jmaa.2011.03.065 Google Scholar
[22] D”Aniello, E., Darji, U. B., and Steele, T. H., Ubiquity of odometers in topological dynamical systems. Topology Appl. 156(2008), no. 2, 240– 245. http://dx.doi.org/10.1016/j.topol.2008.07.003 Google Scholar
[23] Dinaburg, E. I., A connection between various entropy characterizations of dynamical systems. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 35(1971), 324– 366.Google Scholar
[24] Glasner, E. and Weiss, B., The topological Rohlin property and topological entropy. Amer. J. Math. 123(2001), no. 6, 1055– 1070. http://dx.doi.org/10.1353/ajm.2001.0039 Google Scholar
[25] Guirao, J. L. G., Kwietniak, D., Lampart, M., Oprocha, P., and Peris, A., Chaos on hyperspaces. Nonlinear Anal. 71(2009), no. 1– 2, 1– 8. http://dx.doi.org/10.1016/j.na.2008.10.055 Google Scholar
[26] Hochman, M., Genericity in topological dynamics. Ergodic Theory Dynam. Systems 28(2008), no. 1, 125– 165.Google Scholar
[27] Illanes, A. and Nadler, S. B. Jr., Hyperspaces. Fundamentals and recent advances. Monographs and Textbooks in Pure and Applied Mathematics, 216, Marcel Dekker, Inc., New York, 1999.Google Scholar
[28] Kechris, A. S. and Rosendal, C., Turbulence, amalgamation, and generic automorphisms of homogeneous structures. Proc. Lond. Math. Soc. (3) 94(2007), no. 2, 302– 350.http://dx.doi.org/10.1112/plms/pdl007 Google Scholar
[29] Lehning, H., Dynamics of typical continuous functions. Proc. Amer. Math. Soc. 123(1995), no. 6, 1703– 1707. http://dx.doi.org/10.1090/S0002-9939-1995-1239798-X Google Scholar
[30] Li, T. Y. and Yorke, J. A., Period three implies chaos. Amer. Math. Monthly 82(1975), no. 10, 985– 992.http://dx.doi.org/10.2307/2318254 Google Scholar
[31] Oprocha, P., Distributional chaos revisited. Trans. Amer. Math. Soc. 361(2009), no. 9, 4901– 4925.http://dx.doi.org/10.1090/S0002-9947-09-04810-7 Google Scholar
[32] Pilyugin, S. Yu., The space of dynamical systems with the C0topology. Lecture Notes in Mathematics, 1571, Springer-Verlag, Berlin, 1994.Google Scholar
[33] Schweizer, B. and Smítal, J., Measures of chaos and a spectral decomposition of dynamical systems on the interval. Trans. Amer. Math. Soc. 344(1994), no. 2, 737– 754.http://dx.doi.org/10.1090/S0002-9947-1994-1227094-X Google Scholar
[34] Simon, K., On the periodic points of a typical continuous function. Proc. Amer. Math. Soc. 105(1989), no. 1, 244– 249.http://dx.doi.org/10.1090/S0002-9939-1989-0929418-0 Google Scholar
[35] Yano, K., A remark on the topological entropy of homeomorphisms. Invent.Math. 59(1980), no. 3, 215– 220.http://dx.doi.org/10.1007/BF01453235 Google Scholar