# American Institute of Mathematical Sciences

July  2013, 33(7): 3135-3152. doi: 10.3934/dcds.2013.33.3135

## Entropy and exact Devaney chaos on totally regular continua

 1 Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica

Received  March 2012 Revised  May 2012 Published  January 2013

We study topological entropy of exactly Devaney chaotic maps on totally regular continua, i.e. on (topologically) rectifiable curves. After introducing the so-called $P$-Lipschitz maps (where $P$ is a finite invariant set) we give an upper bound for their topological entropy. We prove that if a non-degenerate totally regular continuum $X$ contains a free arc which does not disconnect $X$ or if $X$ contains arbitrarily large generalized stars then $X$ admits an exactly Devaney chaotic map with arbitrarily small entropy. A possible application for further study of the best lower bounds of topological entropies of transitive/Devaney chaotic maps is indicated.
Citation: Vladimír Špitalský. Entropy and exact Devaney chaos on totally regular continua. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3135-3152. doi: 10.3934/dcds.2013.33.3135
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##### References:
 [1] Topology, 36 (1997), 519-532. doi: 10.1016/0040-9383(95)00070-4.  Google Scholar [2] Trans. Amer. Math. Soc., 351 (1999), 1551-1573. doi: 10.1090/S0002-9947-99-02077-2.  Google Scholar [3] $2^{nd}$ edition, Advanced Series in Nonlinear Dynamics 5, World Scientific, Singapore, 2000.  Google Scholar [4] J. Math. Anal. Appl., 232 (1999), 359-375. doi: 10.1006/jmaa.1999.6277.  Google Scholar [5] Topology, 40 (2001), 551-569. doi: 10.1016/S0040-9383(99)00074-9.  Google Scholar [6] Topology Appl., 133 (2003), 225-239. doi: 10.1016/S0166-8641(03)00090-7.  Google Scholar [7] Bull. Amer. Math. Soc., 55 (1949), 1101-1110.  Google Scholar [8] Russian Math. Surveys, 37 (1982), 203-204.  Google Scholar [9] (Russian), Differential-Difference Equations and Problems of Mathematical Physics, 3-9, 131, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, (1984).  Google Scholar [10] Russ. Math. Surv., 42 (1987), 165-166.  Google Scholar [11] Trans. Amer. Math. Soc., 153 (1971), 401-414; Erratum: Trans. Amer. Math. Soc. 181 (1973), 509-510.  Google Scholar [12] Houston J. Math., 18 (1992), 319-327.  Google Scholar [13] (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366.  Google Scholar [14] M. Dirbák, L'. Snoha and V. Špitalský, Minimality, transitivity, mixing and topological entropy on spaces with a free interval,, Ergodic Theory Dynam. Systems, ().  doi: 10.1017/S0143385712000442.  Google Scholar [15] Cambridge University Press, Cambridge, 1986.  Google Scholar [16] Springer-Verlag New York Inc., New York, 1969.  Google Scholar [17] Proc. London Math. Soc., 64 (1992), 449-486. doi: 10.1112/plms/s3-64.3.449.  Google Scholar [18] G. Harańczyk, D. Kwietniak and P. Oprocha, Topological structure and entropy of mixing graph maps,, preprint, ().   Google Scholar [19] Cambridge University Press, Cambridge, 1995.  Google Scholar [20] Discrete Contin. Dyn. Syst., 30 (2011), 767-777. doi: 10.3934/dcds.2011.30.767.  Google Scholar [21] Academic Press and PWN, Warszawa, 1968.  Google Scholar [22] Qual. Theory Dyn. Syst., 6 (2005), 169-179. doi: 10.1007/BF02972670.  Google Scholar [23] Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420026535.  Google Scholar [24] Cambridge University Press, Cambridge, 1995.  Google Scholar [25] Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992.  Google Scholar [26] Fund. Math., 133 (1989), 125-134.  Google Scholar [27] S. Ruette, Chaos for continuous interval maps - a survey of relationship between the various sorts of chaos,, preprint, ().   Google Scholar [28] V. Špitalský, Length-expanding Lipschitz maps on totally regular continua,, preprint, ().   Google Scholar [29] American Mathematical Society, New York, 1942.  Google Scholar [30] Ergodic Theory Dynam. Systems, 20 (2000), 289-314. doi: 10.1017/S0143385700000134.  Google Scholar
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