American Institute of Mathematical Sciences

Advanced
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, (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.  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.  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.  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).   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.  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.   Google Scholar

[8]

J. Bobok, The topological entropy versus level sets for interval maps. II, Studia Math., 166 (2005), 11.  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.  doi: 10.1017/S0143385704000574.  Google Scholar

[11]

A. Bruckner, Differentiation of Real Functions,, Second edition, (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.  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.  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.  doi: 10.1090/S0002-9939-09-10162-4.  Google Scholar

[15]

P. Kůrka, Topological and Symbolic Dynamics,, Cours Spécialisés [Specialized Courses], (2003).   Google Scholar

[16]

R. Mañé, Ergodic Theory and Differentiable Dynamics,, Springer, (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.   Google Scholar

[18]

C. Mouron, Entropy of shift maps of the pseudo-arc,, Topology Appl., 159 (2012), 34.  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, (1982).   Google Scholar

show all references

References:
[1]

Ll. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, Second edition, (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.  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.  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.  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).   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.  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.   Google Scholar

[8]

J. Bobok, The topological entropy versus level sets for interval maps. II, Studia Math., 166 (2005), 11.  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.  doi: 10.1017/S0143385704000574.  Google Scholar

[11]

A. Bruckner, Differentiation of Real Functions,, Second edition, (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.  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.  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.  doi: 10.1090/S0002-9939-09-10162-4.  Google Scholar

[15]

P. Kůrka, Topological and Symbolic Dynamics,, Cours Spécialisés [Specialized Courses], (2003).   Google Scholar

[16]

R. Mañé, Ergodic Theory and Differentiable Dynamics,, Springer, (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.   Google Scholar

[18]

C. Mouron, Entropy of shift maps of the pseudo-arc,, Topology Appl., 159 (2012), 34.  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, (1982).   Google Scholar

[1]

David Burguet. Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 873-899. doi: 10.3934/dcds.2010.26.873
[2]

Asaf Katz. On mixing and sparse ergodic theorems. Journal of Modern Dynamics, 2021, 17: 1-32. doi: 10.3934/jmd.2021001
[3]

Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295
[4]

Cristina Lizana, Vilton Pinheiro, Paulo Varandas. Contribution to the ergodic theory of robustly transitive maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 353-365. doi: 10.3934/dcds.2015.35.353
[5]

Vladimír Špitalský. Transitive dendrite map with infinite decomposition ideal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 771-792. doi: 10.3934/dcds.2015.35.771
[6]

Mykola Matviichuk, Damoon Robatian. Chain transitive induced interval maps on continua. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 741-755. doi: 10.3934/dcds.2015.35.741
[7]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[8]

Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451
[9]

François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[10]

Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079
[11]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[12]

Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012
[13]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181
[14]

Yuanhong Chen, Chao Ma, Jun Wu. Moving recurrent properties for the doubling map on the unit interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 2969-2979. doi: 10.3934/dcds.2016.36.2969
[15]

Wen Huang, Leiye Xu, Shengnan Xu. Ergodic measures of intermediate entropy for affine transformations of nilmanifolds. Electronic Research Archive, , () : -. doi: 10.3934/era.2021015
[16]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
[17]

James P. Kelly, Kevin McGoff. Entropy conjugacy for Markov multi-maps of the interval. Discrete & Continuous Dynamical Systems - A, 2021, 41 (5) : 2071-2094. doi: 10.3934/dcds.2020353
[18]

Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289
[19]

Jérôme Buzzi, Sylvie Ruette. Large entropy implies existence of a maximal entropy measure for interval maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 673-688. doi: 10.3934/dcds.2006.14.673
[20]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences