Entropy and exact Devaney chaos on totally regular continua

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

Entropy and exact Devaney chaos on totally regular continua

Vladimír Špitalský ^1,

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

Abstract
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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.

Keywords: topological entropy, Lipschitz map, totally regular continuum, Exact Devaney chaos, continuum of finite length, rectifiable curve..

Mathematics Subject Classification: Primary: 37B05, 37B20, 37B40; Secondary: 54H2.

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, (). 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, ().
[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, ().
[28]	V. Špitalský, Length-expanding Lipschitz maps on totally regular continua,, preprint, ().
[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, (). 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, ().
[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, ().
[28]	V. Špitalský, Length-expanding Lipschitz maps on totally regular continua,, preprint, ().
[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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