Topological mixing, knot points and bounds of topological entropy

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December 2015, 20(10): 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

Topological mixing, knot points and bounds of topological entropy

Piotr Oprocha ^1, and Paweł Potorski ^2,

1.	Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków
2.	AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland

Received December 2014 Revised March 2015 Published September 2015

Abstract
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In the paper we provide exact lower bounds of topological entropy in the class of transitive and mixing maps preserving the Lebesgue measure which are nowhere monotone (with dense knot points).

Keywords: transitive, knot point, mixing, ergodic., Entropy, interval map.

Mathematics Subject Classification: Primary: 37B40; Secondary: 37E05, 37A0.

Citation: Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547

References:

[1]	Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Second edition, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/4205.
[2]	M. Barge and J. Martin, Dense periodicity on the interval, Proc. Amer. Math. Soc., 94 (1985), 731-735. doi: 10.1090/S0002-9939-1985-0792293-8.
[3]	M. Barge and J. Martin, Dense orbits on the interval, Michigan Math. J., 34 (1987), 3-11. doi: 10.1307/mmj/1029003477.
[4]	A. Barrio Blaya and V. Jiménez López, On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps, Discrete Contin. Dyn. Syst., 32 (2012), 433-466. doi: 10.3934/dcds.2012.32.433.
[5]	L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513, Springer, Berlin, 1992.
[6]	L. Block and E. M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, Trans. Amer. Math. Soc., 300 (1987), 297-306. doi: 10.1090/S0002-9947-1987-0871677-X.
[7]	J. Bobok, Strictly ergodic patterns and entropy for interval maps, Acta Math. Univ. Comenianae, 72 (2003), 111-118.
[8]	J. Bobok, The topological entropy versus level sets for interval maps. II Studia Math., 166 (2005), 11-27. doi: 10.4064/sm166-1-2.
[9]	J. Bobok and M. Soukenka, Irreducibility, infinite level sets, and small entropy, Real Analysis Exchange, 36 (2010/2011), 449-461.
[10]	J. Bobok and Z. Nitecki, The topological entropy of $m$-fold maps, Ergod. Th. Dynam. Sys., 25 (2005), 375-401. doi: 10.1017/S0143385704000574.
[11]	A. Bruckner, Differentiation of Real Functions, Second edition, CRM Monograph Series, 5, American Mathematical Society, Providence, RI, 1994.
[12]	G. Harańczyk and D. Kwietniak, When lower entropy implies stronger Devanay chaos, Proceedings of the American Mathematical Society, 137 (2009), 2063-2073. doi: 10.1090/S0002-9939-08-09756-6.
[13]	G. W. Henderson, The pseudo-arc as an inverse limit with one binding map, Duke Math. J., 31 (1964), 421-425. doi: 10.1215/S0012-7094-64-03140-0.
[14]	P. Kościelniak and P. Oprocha, Shadowing, entropy and a homeomorphism of the pseudoarc, Proc. Amer. Math. Soc., 138 (2010), 1047-1057. doi: 10.1090/S0002-9939-09-10162-4.
[15]	P. Kůrka, Topological and Symbolic Dynamics, Cours Spécialisés [Specialized Courses], 11, Société Mathématique de France, Paris, 2003.
[16]	R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.
[17]	M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.
[18]	C. Mouron, Entropy of shift maps of the pseudo-arc, Topology Appl., 159 (2012), 34-39. doi: 10.1016/j.topol.2011.07.014.
[19]	S. Ruette, Chaos for continuous interval maps, preprint, 2003. Available from: http://www.math.u-psud.fr/ ruette/publications.html.
[20]	P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.

show all references

References:

[1]	Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Second edition, Advanced Series in Nonlinear Dynamics, 5, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/4205.
[2]	M. Barge and J. Martin, Dense periodicity on the interval, Proc. Amer. Math. Soc., 94 (1985), 731-735. doi: 10.1090/S0002-9939-1985-0792293-8.
[3]	M. Barge and J. Martin, Dense orbits on the interval, Michigan Math. J., 34 (1987), 3-11. doi: 10.1307/mmj/1029003477.
[4]	A. Barrio Blaya and V. Jiménez López, On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps, Discrete Contin. Dyn. Syst., 32 (2012), 433-466. doi: 10.3934/dcds.2012.32.433.
[5]	L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513, Springer, Berlin, 1992.
[6]	L. Block and E. M. Coven, Topological conjugacy and transitivity for a class of piecewise monotone maps of the interval, Trans. Amer. Math. Soc., 300 (1987), 297-306. doi: 10.1090/S0002-9947-1987-0871677-X.
[7]	J. Bobok, Strictly ergodic patterns and entropy for interval maps, Acta Math. Univ. Comenianae, 72 (2003), 111-118.
[8]	J. Bobok, The topological entropy versus level sets for interval maps. II Studia Math., 166 (2005), 11-27. doi: 10.4064/sm166-1-2.
[9]	J. Bobok and M. Soukenka, Irreducibility, infinite level sets, and small entropy, Real Analysis Exchange, 36 (2010/2011), 449-461.
[10]	J. Bobok and Z. Nitecki, The topological entropy of $m$-fold maps, Ergod. Th. Dynam. Sys., 25 (2005), 375-401. doi: 10.1017/S0143385704000574.
[11]	A. Bruckner, Differentiation of Real Functions, Second edition, CRM Monograph Series, 5, American Mathematical Society, Providence, RI, 1994.
[12]	G. Harańczyk and D. Kwietniak, When lower entropy implies stronger Devanay chaos, Proceedings of the American Mathematical Society, 137 (2009), 2063-2073. doi: 10.1090/S0002-9939-08-09756-6.
[13]	G. W. Henderson, The pseudo-arc as an inverse limit with one binding map, Duke Math. J., 31 (1964), 421-425. doi: 10.1215/S0012-7094-64-03140-0.
[14]	P. Kościelniak and P. Oprocha, Shadowing, entropy and a homeomorphism of the pseudoarc, Proc. Amer. Math. Soc., 138 (2010), 1047-1057. doi: 10.1090/S0002-9939-09-10162-4.
[15]	P. Kůrka, Topological and Symbolic Dynamics, Cours Spécialisés [Specialized Courses], 11, Société Mathématique de France, Paris, 2003.
[16]	R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.
[17]	M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.
[18]	C. Mouron, Entropy of shift maps of the pseudo-arc, Topology Appl., 159 (2012), 34-39. doi: 10.1016/j.topol.2011.07.014.
[19]	S. Ruette, Chaos for continuous interval maps, preprint, 2003. Available from: http://www.math.u-psud.fr/ ruette/publications.html.
[20]	P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.

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