Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T12:16:48.370Z Has data issue: false hasContentIssue false
  • English
  • Français

Topological entropy of transitive dendrite maps

Published online by Cambridge University Press:  14 November 2013

VLADIMÍR ŠPITALSKÝ
VLADIMÍR ŠPITALSKÝ*
Affiliation:
Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia email vladimir.spitalsky@umb.sk
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that every dendrite $X$ satisfying the condition that no subtree of $X$ contains all free arcs admits a transitive, even exactly Devaney chaotic map with arbitrarily small entropy. This gives a partial answer to a question of Baldwin from 2001.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 4 , June 2015 , pp. 1289 - 1314
Copyright
© Cambridge University Press, 2013 

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

Adler, R. L., Konheim, A. G. and McAndrew, M. H.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 309–319.CrossRefGoogle Scholar
Acosta, G., Eslami, P. and Oversteegen, L. G.. On open maps between dendrites. Houston J. Math. 33 (3) (2007), 753–770.Google Scholar
Alsedà, L., Baldwin, S., Llibre, J. and Misiurewicz, M.. Entropy of transitive tree maps. Topology 36 (2) (1997), 519–532.CrossRefGoogle Scholar
Alsedà, L., Kolyada, S., Llibre, J. and Snoha, L’.. Entropy and periodic points for transitive maps. Trans. Amer. Math. Soc. 351 (4) (1999), 1551–1573.CrossRefGoogle Scholar
Alsedà, L., del Río, M. A. and Rodríguez, J. A.. A splitting theorem for transitive maps. J. Math. Anal. Appl. 232 (2) (1999), 359–375.CrossRefGoogle Scholar
Baillif, M. and de Carvalho, A.. Piecewise linear model for tree maps. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11 (12) (2001), 3163–3169.CrossRefGoogle Scholar
Baldwin, S.. Entropy estimates for transitive maps on trees. Topology 40 (3) (2001), 551–569.CrossRefGoogle Scholar
Baldwin, S.. Continuous itinerary functions and dendrite maps. Topology Appl. 154 (16) (2007), 2889–2938.CrossRefGoogle Scholar
Bing, R. H.. Partitioning a set. Bull. Amer. Math. Soc. 55 (1949), 1101–1110.CrossRefGoogle Scholar
Block, L., Guckenheimer, J., Misiurewicz, M. and Young, L. S.. Periodic points and topological entropy of one dimentional maps. Global Theory of Dynamical Systems (Lecture Notes in Mathematics, 819). Springer, New York, 1980, pp. 18–34.CrossRefGoogle Scholar
Blokh, A.. On sensitive mappings of the interval. Russian Math. Surveys 37 (1982), 203–204.CrossRefGoogle Scholar
Blokh, A.. On the connection between entropy and transitivity for one-dimensional mappings. Russian Math. Surveys 42 (5) (1987), 165–166.CrossRefGoogle Scholar
Dirbák, M., Snoha, L’. and Špitalský, V.. Minimality, transitivity, mixing and topological entropy on spaces with a free interval. Ergod. Th. & Dynam. Sys. (2012), doi:10.1017/S0143385712000442.Google Scholar
Eilenberg, S. and Harrold, O. G.. Continua of finite linear measure I. Amer. J. Math. 65 (1943), 137–146.CrossRefGoogle Scholar
Harańczyk, G., Kwietniak, D. and Oprocha, P.. Topological structure and entropy of mixing graph maps. Ergod. Th. & Dynam. Sys. (2013), doi:10.1017/etds.2013.6.Google Scholar
Illanes, A.. A characterization of dendrites with the periodic-recurrent property. Topology Proc. 23 (1998), 221–235.Google Scholar
Kato, H.. The depth of centres of maps on dendrites. J. Aust. Math. Soc. Ser. A 64 (1) (1998), 44–53.CrossRefGoogle Scholar
Makhrova, E. N.. Homoclinic points and topological entropy of a continuous mapping of a dendrite. Sovrem. Mat. Prilozh. 54(2) (2008), 79–86 (Trudy Mezhdunarodnoi Konferentsii po Dinamicheskim Sistemam i Differentsialnym Uravneniyam) (in Russian); Engl. transl. J. Math. Sci. (N.Y.) 158(2) (2009), 241–248.Google Scholar
Nadler, S. B.. Continuum Theory. An Introduction (Monographs and Textbooks in Pure and Applied Mathematics, 158). Marcel Dekker, New York, 1992.Google Scholar
Naghmouchi, I.. Dynamical properties of monotone dendrite maps. Topology Appl. 159 (1) (2012), 144–149.CrossRefGoogle Scholar
Seneta, E.. Non-negative Matrices and Markov Chains (Springer Series in Statistics). Springer, New York, 2006, revised reprint of the second (1981) edition.Google Scholar
Špitalský, V.. Length-expanding Lipschitz maps on totally regular continua. J. Math. Anal. Appl. to appear. Preprint, 2012, arXiv:1203.2352 [math.DS].Google Scholar
Špitalský, V.. Entropy and exact Devaney chaos on totally regular continua. Discrete Contin. Dyn. Syst. 33 (7) (2013), 3135–3152.CrossRefGoogle Scholar
Shi, E., Wang, S. and Zhou, L.. Minimal group actions on dendrites. Proc. Amer. Math. Soc. 138 (1) (2010), 217–223.CrossRefGoogle Scholar
Ye, X.. Topological entropy of transitive maps of a tree. Ergod. Th. & Dynam. Sys. 20 (1) (2000), 289–314.CrossRefGoogle Scholar