# 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 and Continuous Dynamical Systems, 2013, 33 (7) : 3135-3152. doi: 10.3934/dcds.2013.33.3135
##### References:
 [1] L. Alsedà, S. Baldwin, J. Llibre and M. Misiurewicz, Entropy of transitive tree maps, Topology, 36 (1997), 519-532. doi: 10.1016/0040-9383(95)00070-4. [2] L. Alsedà, S. Kolyada, J. Llibre and L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc., 351 (1999), 1551-1573. doi: 10.1090/S0002-9947-99-02077-2. [3] Ll. Alsedà, J. Llibre and M. Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One," $2^{nd}$ edition, Advanced Series in Nonlinear Dynamics 5, World Scientific, Singapore, 2000. [4] L. Alsedà, M. A. del Río and J. A. Rodríguez, A splitting theorem for transitive maps, J. Math. Anal. Appl., 232 (1999), 359-375. doi: 10.1006/jmaa.1999.6277. [5] S. Baldwin, Entropy estimates for transitive maps on trees, Topology, 40 (2001), 551-569. doi: 10.1016/S0040-9383(99)00074-9. [6] F. Balibrea and L'. Snoha, Topological entropy of Devaney chaotic maps, Topology Appl., 133 (2003), 225-239. doi: 10.1016/S0166-8641(03)00090-7. [7] R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc., 55 (1949), 1101-1110. [8] A. Blokh, On sensitive mappings of the interval, Russian Math. Surveys, 37 (1982), 203-204. [9] A. Blokh, On transitive mappings of one-dimensional branched manifolds, (Russian), Differential-Difference Equations and Problems of Mathematical Physics, 3-9, 131, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, (1984). [10] A. Blokh, On the connection between entropy and transitivity for one-dimensional mappings, Russ. Math. Surv., 42 (1987), 165-166. [11] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414; Erratum: Trans. Amer. Math. Soc. 181 (1973), 509-510. [12] R. D. Buskirk, J. Nikiel and E. D. Tymchatyn, Totally regular curves as inverse limits, Houston J. Math., 18 (1992), 319-327. [13] E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366. [14] M. Dirbák, L'. Snoha and V. Špitalský, Minimality, transitivity, mixing and topological entropy on spaces with a free interval, Ergodic Theory Dynam. Systems, available on CJO2012. doi: 10.1017/S0143385712000442. [15] K. J. Falconer, "The Geometry of Fractal Sets," Cambridge University Press, Cambridge, 1986. [16] H. Federer, "Geometric Measure Theory," Springer-Verlag New York Inc., New York, 1969. [17] D. H. Fremlin, Spaces of finite length, Proc. London Math. Soc., 64 (1992), 449-486. doi: 10.1112/plms/s3-64.3.449. [18] G. Harańczyk, D. Kwietniak and P. Oprocha, Topological structure and entropy of mixing graph maps, preprint, arXiv:1111.0566. [19] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995. [20] S. Kolyada and M. Matviichuk, On extensions of transitive maps, Discrete Contin. Dyn. Syst., 30 (2011), 767-777. doi: 10.3934/dcds.2011.30.767. [21] K. Kuratowski, "Topology, vol. 2," Academic Press and PWN, Warszawa, 1968. [22] D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst., 6 (2005), 169-179. doi: 10.1007/BF02972670. [23] S. Macías, "Topics on Continua," Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420026535. [24] P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge University Press, Cambridge, 1995. [25] S. B. Nadler, "Continuum Theory. An Introduction," Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992. [26] J. Nikiel, Locally connected curves viewed as inverse limits, Fund. Math., 133 (1989), 125-134. [27] S. Ruette, Chaos for continuous interval maps - a survey of relationship between the various sorts of chaos, preprint, available at http://www.math.u-psud.fr/~ruette/articles/chaos-int.pdf. [28] V. Špitalský, Length-expanding Lipschitz maps on totally regular continua, preprint, arXiv:1203.2352. [29] G. T. Whyburn, "Analytic Topology," American Mathematical Society, New York, 1942. [30] X. Ye, Topological entropy of transitive maps of a tree, Ergodic Theory Dynam. Systems, 20 (2000), 289-314. doi: 10.1017/S0143385700000134.

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##### References:
 [1] L. Alsedà, S. Baldwin, J. Llibre and M. Misiurewicz, Entropy of transitive tree maps, Topology, 36 (1997), 519-532. doi: 10.1016/0040-9383(95)00070-4. [2] L. Alsedà, S. Kolyada, J. Llibre and L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc., 351 (1999), 1551-1573. doi: 10.1090/S0002-9947-99-02077-2. [3] Ll. Alsedà, J. Llibre and M. Misiurewicz, "Combinatorial Dynamics and Entropy in Dimension One," $2^{nd}$ edition, Advanced Series in Nonlinear Dynamics 5, World Scientific, Singapore, 2000. [4] L. Alsedà, M. A. del Río and J. A. Rodríguez, A splitting theorem for transitive maps, J. Math. Anal. Appl., 232 (1999), 359-375. doi: 10.1006/jmaa.1999.6277. [5] S. Baldwin, Entropy estimates for transitive maps on trees, Topology, 40 (2001), 551-569. doi: 10.1016/S0040-9383(99)00074-9. [6] F. Balibrea and L'. Snoha, Topological entropy of Devaney chaotic maps, Topology Appl., 133 (2003), 225-239. doi: 10.1016/S0166-8641(03)00090-7. [7] R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc., 55 (1949), 1101-1110. [8] A. Blokh, On sensitive mappings of the interval, Russian Math. Surveys, 37 (1982), 203-204. [9] A. Blokh, On transitive mappings of one-dimensional branched manifolds, (Russian), Differential-Difference Equations and Problems of Mathematical Physics, 3-9, 131, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, (1984). [10] A. Blokh, On the connection between entropy and transitivity for one-dimensional mappings, Russ. Math. Surv., 42 (1987), 165-166. [11] R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414; Erratum: Trans. Amer. Math. Soc. 181 (1973), 509-510. [12] R. D. Buskirk, J. Nikiel and E. D. Tymchatyn, Totally regular curves as inverse limits, Houston J. Math., 18 (1992), 319-327. [13] E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 35 (1971), 324-366. [14] M. Dirbák, L'. Snoha and V. Špitalský, Minimality, transitivity, mixing and topological entropy on spaces with a free interval, Ergodic Theory Dynam. Systems, available on CJO2012. doi: 10.1017/S0143385712000442. [15] K. J. Falconer, "The Geometry of Fractal Sets," Cambridge University Press, Cambridge, 1986. [16] H. Federer, "Geometric Measure Theory," Springer-Verlag New York Inc., New York, 1969. [17] D. H. Fremlin, Spaces of finite length, Proc. London Math. Soc., 64 (1992), 449-486. doi: 10.1112/plms/s3-64.3.449. [18] G. Harańczyk, D. Kwietniak and P. Oprocha, Topological structure and entropy of mixing graph maps, preprint, arXiv:1111.0566. [19] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995. [20] S. Kolyada and M. Matviichuk, On extensions of transitive maps, Discrete Contin. Dyn. Syst., 30 (2011), 767-777. doi: 10.3934/dcds.2011.30.767. [21] K. Kuratowski, "Topology, vol. 2," Academic Press and PWN, Warszawa, 1968. [22] D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst., 6 (2005), 169-179. doi: 10.1007/BF02972670. [23] S. Macías, "Topics on Continua," Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420026535. [24] P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge University Press, Cambridge, 1995. [25] S. B. Nadler, "Continuum Theory. An Introduction," Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992. [26] J. Nikiel, Locally connected curves viewed as inverse limits, Fund. Math., 133 (1989), 125-134. [27] S. Ruette, Chaos for continuous interval maps - a survey of relationship between the various sorts of chaos, preprint, available at http://www.math.u-psud.fr/~ruette/articles/chaos-int.pdf. [28] V. Špitalský, Length-expanding Lipschitz maps on totally regular continua, preprint, arXiv:1203.2352. [29] G. T. Whyburn, "Analytic Topology," American Mathematical Society, New York, 1942. [30] X. Ye, Topological entropy of transitive maps of a tree, Ergodic Theory Dynam. Systems, 20 (2000), 289-314. doi: 10.1017/S0143385700000134.
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