When lower entropy implies stronger Devaney chaos

Harańczyk, Grzegorz; Kwietniak, Dominik

doi:10.1090/S0002-9939-08-09756-6

When lower entropy implies stronger Devaney chaos
HTML articles powered by AMS MathViewer

by Grzegorz Harańczyk and Dominik Kwietniak PDF

Proc. Amer. Math. Soc. 137 (2009), 2063-2073 Request permission

Abstract:

It is proved that the infimum of the topological entropy of continuous topologically exact interval (circle) maps is strictly smaller than the infimum of the topological entropy of continuous interval (circle) maps, which are topologically mixing, but not exact. Interpreting this result in terms of popular notions of chaos, one may say that on the interval (circle) lower entropy implies stronger Devaney chaos. Moreover, the infimum of the entropy of mixing circle maps is computed. These theorems may be considered as a completion of some results of Alsedà, Kolyada, Llibre, and Snoha (1999).

References

Ll. Alsedà, S. Kolyada, J. Llibre, and Ľ. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc. 351 (1999), no. 4, 1551–1573. MR 1451592, DOI 10.1090/S0002-9947-99-02077-2
Lluis Alsedà, Stewart Baldwin, Jaume Llibre, and MichałMisiurewicz, Entropy of transitive tree maps, Topology 36 (1997), no. 2, 519–532. MR 1415604, DOI 10.1016/0040-9383(95)00070-4
Lluís Alsedà, Jaume Llibre, and MichałMisiurewicz, Combinatorial dynamics and entropy in dimension one, 2nd ed., Advanced Series in Nonlinear Dynamics, vol. 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR 1807264, DOI 10.1142/4205
J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney’s definition of chaos, Amer. Math. Monthly 99 (1992), no. 4, 332–334. MR 1157223, DOI 10.2307/2324899
Marcy Barge and Joe Martin, Chaos, periodicity, and snakelike continua, Trans. Amer. Math. Soc. 289 (1985), no. 1, 355–365. MR 779069, DOI 10.1090/S0002-9947-1985-0779069-7
Marcy Barge and Joe Martin, Dense periodicity on the interval, Proc. Amer. Math. Soc. 94 (1985), no. 4, 731–735. MR 792293, DOI 10.1090/S0002-9939-1985-0792293-8
Marcy Barge and Joe Martin, Dense orbits on the interval, Michigan Math. J. 34 (1987), no. 1, 3–11. MR 873014, DOI 10.1307/mmj/1029003477
A. M. Blokh, Sensitive mappings of an interval, Uspekhi Mat. Nauk 37 (1982), no. 2(224), 189–190 (Russian). MR 650765
Alexander M. Blokh, The “spectral” decomposition for one-dimensional maps, Dynamics reported, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 4, Springer, Berlin, 1995, pp. 1–59. MR 1346496
Ethan M. Coven and Irene Mulvey, Transitivity and the centre for maps of the circle, Ergodic Theory Dynam. Systems 6 (1986), no. 1, 1–8. MR 837972, DOI 10.1017/S0143385700003254
Robert L. Devaney, An introduction to chaotic dynamical systems, Studies in Nonlinearity, Westview Press, Boulder, CO, 2003. Reprint of the second (1989) edition. MR 1979140
Sergiĭ Kolyada and Ľubomír Snoha, Some aspects of topological transitivity—a survey, Iteration theory (ECIT 94) (Opava), Grazer Math. Ber., vol. 334, Karl-Franzens-Univ. Graz, Graz, 1997, pp. 3–35. MR 1644768
Dominik Kwietniak and MichałMisiurewicz, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst. 6 (2005), no. 1, 169–179. MR 2273492, DOI 10.1007/BF02972670
Sylvie Ruette, Chaos for continuous interval maps, preprint, available at http://www.math.u-psud.fr/~ruette/publications.html
Benjamin Weiss, Topological transitivity and ergodic measures, Math. Systems Theory 5 (1971), 71–75. MR 296928, DOI 10.1007/BF01691469