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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. Baldwin, Entropy estimates for transitive maps on trees, Topology, 40 (2001), 551-569. doi: 10.1016/S0040-9383(99)00074-9.  Google Scholar

[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.  Google Scholar

[7]

R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc., 55 (1949), 1101-1110.  Google Scholar

[8]

A. Blokh, On sensitive mappings of the interval, Russian Math. Surveys, 37 (1982), 203-204.  Google Scholar

[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).  Google Scholar

[10]

A. Blokh, On the connection between entropy and transitivity for one-dimensional mappings, Russ. Math. Surv., 42 (1987), 165-166.  Google Scholar

[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.  Google Scholar

[12]

R. D. Buskirk, J. Nikiel and E. D. Tymchatyn, Totally regular curves as inverse limits, Houston J. Math., 18 (1992), 319-327.  Google Scholar

[13]

E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, (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]

K. J. Falconer, "The Geometry of Fractal Sets," Cambridge University Press, Cambridge, 1986.  Google Scholar

[16]

H. Federer, "Geometric Measure Theory," Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[17]

D. H. Fremlin, Spaces of finite length, 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]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.  Google Scholar

[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.  Google Scholar

[21]

K. Kuratowski, "Topology, vol. 2," Academic Press and PWN, Warszawa, 1968.  Google Scholar

[22]

D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst., 6 (2005), 169-179. doi: 10.1007/BF02972670.  Google Scholar

[23]

S. Macías, "Topics on Continua," Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420026535.  Google Scholar

[24]

P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge University Press, Cambridge, 1995.  Google Scholar

[25]

S. B. Nadler, "Continuum Theory. An Introduction," Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992.  Google Scholar

[26]

J. Nikiel, Locally connected curves viewed as inverse limits, 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]

G. T. Whyburn, "Analytic Topology," American Mathematical Society, New York, 1942.  Google Scholar

[30]

X. Ye, Topological entropy of transitive maps of a tree, Ergodic Theory Dynam. Systems, 20 (2000), 289-314. doi: 10.1017/S0143385700000134.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

S. Baldwin, Entropy estimates for transitive maps on trees, Topology, 40 (2001), 551-569. doi: 10.1016/S0040-9383(99)00074-9.  Google Scholar

[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.  Google Scholar

[7]

R. H. Bing, Partitioning a set, Bull. Amer. Math. Soc., 55 (1949), 1101-1110.  Google Scholar

[8]

A. Blokh, On sensitive mappings of the interval, Russian Math. Surveys, 37 (1982), 203-204.  Google Scholar

[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).  Google Scholar

[10]

A. Blokh, On the connection between entropy and transitivity for one-dimensional mappings, Russ. Math. Surv., 42 (1987), 165-166.  Google Scholar

[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.  Google Scholar

[12]

R. D. Buskirk, J. Nikiel and E. D. Tymchatyn, Totally regular curves as inverse limits, Houston J. Math., 18 (1992), 319-327.  Google Scholar

[13]

E. I. Dinaburg, A connection between various entropy characterizations of dynamical systems, (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]

K. J. Falconer, "The Geometry of Fractal Sets," Cambridge University Press, Cambridge, 1986.  Google Scholar

[16]

H. Federer, "Geometric Measure Theory," Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[17]

D. H. Fremlin, Spaces of finite length, 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]

A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," Cambridge University Press, Cambridge, 1995.  Google Scholar

[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.  Google Scholar

[21]

K. Kuratowski, "Topology, vol. 2," Academic Press and PWN, Warszawa, 1968.  Google Scholar

[22]

D. Kwietniak and M. Misiurewicz, Exact Devaney chaos and entropy, Qual. Theory Dyn. Syst., 6 (2005), 169-179. doi: 10.1007/BF02972670.  Google Scholar

[23]

S. Macías, "Topics on Continua," Chapman & Hall/CRC, Boca Raton, FL, 2005. doi: 10.1201/9781420026535.  Google Scholar

[24]

P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge University Press, Cambridge, 1995.  Google Scholar

[25]

S. B. Nadler, "Continuum Theory. An Introduction," Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992.  Google Scholar

[26]

J. Nikiel, Locally connected curves viewed as inverse limits, 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]

G. T. Whyburn, "Analytic Topology," American Mathematical Society, New York, 1942.  Google Scholar

[30]

X. Ye, Topological entropy of transitive maps of a tree, Ergodic Theory Dynam. Systems, 20 (2000), 289-314. doi: 10.1017/S0143385700000134.  Google Scholar

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