Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-04T19:21:43.832Z Has data issue: false hasContentIssue false
  • English
  • Français

Some properties of positive entropy maps

Published online by Cambridge University Press:  15 January 2013

A. ARBIETO  and
C. A. MORALES
A. ARBIETO
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970, Rio de Janeiro, Brazil (email: [email protected], [email protected])
C. A. MORALES
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, PO Box 68530, 21945-970, Rio de Janeiro, Brazil (email: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the stable classes for continuous maps on compact metric spaces have measure zero with respect to any ergodic invariant measure with positive entropy. Then, every continuous map with positive topological entropy on a compact metric space has uncountably many stable classes. We also prove that every continuous map with positive topological entropy of a compact metric space cannot be Lyapunov stable on its recurrent set. For homeomorphisms on compact metric spaces we prove that the sets of heteroclinic points, and sinks in the canonical coordinates case, have zero measure with respect to any ergodic invariant measure with positive entropy. These results generalize those of Fedorenko and Smital [Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.)60 (1) (1991), 11–14], Huang and Ye [Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl.117 (3) (2002), 259–272], Reddy [The existence of expansive homeomorphisms on manifolds. Duke Math. J.32 (1965), 627–632], Reddy and Robertson [Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math.17 (4) (1987), 673–681], Sindelarova [A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268], and Zhou [Some equivalent conditions for self-mappings of a circle. Chinese Ann. Math. Ser. A12(suppl.) (1991), 22–27].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 765 - 776
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Aoki, N. and Hiraide, K.. Topological Theory of Dynamical Systems. Recent Advances (North-Holland Mathematical Library, 52). North-Holland Publishing Co., Amsterdam, 1994.Google Scholar
[2]Block, L. S. and Coppel, W. A.. Dynamics in One Dimension (Lecture Notes in Mathematics, 1513). Springer, Berlin, 1992.Google Scholar
[3]Bowen, R.. Topological entropy and Axiom A. Global Analysis (Proceedings of Symposia in Pure Mathematics, XIV, Berkeley, CA, 1968). American Mathematical Society, Providence, RI, 1970, pp. 23–41.Google Scholar
[4]Brin, M. and Katok, A.. On Local Entropy, Geometric Dynamics (Rio de Janeiro, 1981) (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 30–38.Google Scholar
[5]Cadre, B. and Jacob, P.. On pairwise sensitivity. J. Math. Anal. Appl. 309(1) (2005), 375–382.Google Scholar
[6]Fedorenko, V. V. and Smital, J.. Maps of the interval Ljapunov stable on the set of nonwandering points. Acta Math. Univ. Comenian. (N.S.) 60(1) (1991), 11–14.Google Scholar
[7]Huang, W. and Ye, X.. Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topology Appl. 117(3) (2002), 259–272.Google Scholar
[8]Huang, W., Lu, P. and Ye, X.. Measure-theoretical sensitivity and equicontinuity. Israel J. Math. 183 (2011), 233–283.Google Scholar
[9]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995, with a supplementary chapter by A. Katok and L. Mendoza.Google Scholar
[10]Morales, C. A.. On pairwise sensitive homeomorphisms. Preprint, 2011.Google Scholar
[11]Parthasarathy, K. R., Ranga, R. R. and Varadhan, S. R. S.. On the category of indecomposable distributions on topological groups. Trans. Amer. Math. Soc. 102 (1962), 200–217.Google Scholar
[12]Petersen, K.. Ergodic Theory (Cambridge Studies in Advanced Mathematics, 2). Cambridge University Press, Cambridge, 1983.Google Scholar
[13]Reddy, W.. The existence of expansive homeomorphisms on manifolds. Duke Math. J. 32 (1965), 627–632.CrossRefGoogle Scholar
[14]Reddy, W. and Robertson, L. C.. Sources, sinks and saddles for expansive homeomorphisms with canonical coordinates. Rocky Mountain J. Math. 17(4) (1987), 673–681.CrossRefGoogle Scholar
[15]Sindelarova, P.. A counterexample to a statement concerning Lyapunov stability. Acta Math. Univ. Comenian. 70 (2001), 265–268.Google Scholar
[16]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar
[17]Zhou, Z. L.. Some equivalent conditions for self-mappings of a circle. Chin. Ann. Math. Ser. A 12(suppl.) (1991), 22–27.Google Scholar