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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

M. Barge and J. Martin, Dense orbits on the interval, Michigan Math. J., 34 (1987), 3-11. doi: 10.1307/mmj/1029003477.  Google Scholar

[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.  Google Scholar

[5]

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513, Springer, Berlin, 1992.  Google Scholar

[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.  Google Scholar

[7]

J. Bobok, Strictly ergodic patterns and entropy for interval maps, Acta Math. Univ. Comenianae, 72 (2003), 111-118.  Google Scholar

[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.  Google Scholar

[9]

J. Bobok and M. Soukenka, Irreducibility, infinite level sets, and small entropy,, Real Analysis Exchange, 36 (): 449.   Google Scholar

[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.  Google Scholar

[11]

A. Bruckner, Differentiation of Real Functions, Second edition, CRM Monograph Series, 5, American Mathematical Society, Providence, RI, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Kůrka, Topological and Symbolic Dynamics, Cours Spécialisés [Specialized Courses], 11, Société Mathématique de France, Paris, 2003.  Google Scholar

[16]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[17]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.  Google Scholar

[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.  Google Scholar

[19]

S. Ruette, Chaos for continuous interval maps, preprint,, 2003. Available from: , ().   Google Scholar

[20]

P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

M. Barge and J. Martin, Dense orbits on the interval, Michigan Math. J., 34 (1987), 3-11. doi: 10.1307/mmj/1029003477.  Google Scholar

[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.  Google Scholar

[5]

L. S. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes in Mathematics, 1513, Springer, Berlin, 1992.  Google Scholar

[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.  Google Scholar

[7]

J. Bobok, Strictly ergodic patterns and entropy for interval maps, Acta Math. Univ. Comenianae, 72 (2003), 111-118.  Google Scholar

[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.  Google Scholar

[9]

J. Bobok and M. Soukenka, Irreducibility, infinite level sets, and small entropy,, Real Analysis Exchange, 36 (): 449.   Google Scholar

[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.  Google Scholar

[11]

A. Bruckner, Differentiation of Real Functions, Second edition, CRM Monograph Series, 5, American Mathematical Society, Providence, RI, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

P. Kůrka, Topological and Symbolic Dynamics, Cours Spécialisés [Specialized Courses], 11, Société Mathématique de France, Paris, 2003.  Google Scholar

[16]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[17]

M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63.  Google Scholar

[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.  Google Scholar

[19]

S. Ruette, Chaos for continuous interval maps, preprint,, 2003. Available from: , ().   Google Scholar

[20]

P. Walters, An Introduction to Ergodic Theory, Springer, New York, 1982.  Google Scholar

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