American Institute of Mathematical Sciences

Advanced
October  2006, 14(4): 673-688. doi: 10.3934/dcds.2006.14.673

Large entropy implies existence of a maximal entropy measure for interval maps

Jérôme Buzzi 1, and  Sylvie Ruette 2,

1. 

Centre de Mathématiques de l'Ecole Polytechnique, U.M.R. 7640 du C.N.R.S., Ecole Polytechnique, 91128 Palaiseau Cedex
2. 

Laboratoire de Mathématiques, Topologie et Dynamique, Bât. 425, Université Paris-Sud, F-91405 Orsay cedex, France

Received  October 2004 Revised  September 2005 Published  January 2006

We give a new type of sufficient condition for the existence of measures with maximal entropy for an interval map $f$, using some non-uniform hyperbolicity to compensate for a lack of smoothness of $f$. More precisely, if the topological entropy of a $C^1$ interval map is greater than the sum of the local entropy and the entropy of the critical points, then there exists at least one measure with maximal entropy. As a corollary, we obtain that any $C^r$ interval map $f$ such that htop(f)  >  2log || f'||∞ / r possesses measures with maximal entropy.
Keywords: Maximal entropy measure, Markov shift., interval map.
Mathematics Subject Classification: 37E05, 37C40, 37B4.
Citation: 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
[1]

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
[2]

Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192
[3]

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
[4]

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
[5]

Michael Schraudner. Projectional entropy and the electrical wire shift. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 333-346. doi: 10.3934/dcds.2010.26.333
[6]

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
[7]

Domingo González, Gamaliel Blé. Core entropy of polynomials with a critical point of maximal order. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 115-130. doi: 10.3934/dcds.2019005
[8]

Erik M. Bollt, Joseph D. Skufca, Stephen J . McGregor. Control entropy: A complexity measure for nonstationary signals. Mathematical Biosciences & Engineering, 2009, 6 (1) : 1-25. doi: 10.3934/mbe.2009.6.1
[9]

Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159
[10]

Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205
[11]

Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2307-2314. doi: 10.3934/dcds.2014.34.2307
[12]

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
[13]

Aldana M. González Montoro, Ricardo Cao, Christel Faes, Geert Molenberghs, Nelson Espinosa, Javier Cudeiro, Jorge Mariño. Cross nearest-spike interval based method to measure synchrony dynamics. Mathematical Biosciences & Engineering, 2014, 11 (1) : 27-48. doi: 10.3934/mbe.2014.11.27
[14]

Yan Wang, Guanggan Chen. Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3121-3135. doi: 10.3934/cpaa.2019140
[15]

Prof. Dr.rer.nat Widodo. Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 191-208. doi: 10.3934/dcds.2002.8.191
[16]

Boris Kruglikov, Martin Rypdal. A piece-wise affine contracting map with positive entropy. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 393-394. doi: 10.3934/dcds.2006.16.393
[17]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781
[18]

Alejo Barrio Blaya, Víctor Jiménez López. On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 433-466. doi: 10.3934/dcds.2012.32.433
[19]

Huyi Hu, Miaohua Jiang, Yunping Jiang. Infimum of the metric entropy of hyperbolic attractors with respect to the SRB measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 215-234. doi: 10.3934/dcds.2008.22.215
[20]

Manfred Einsiedler, Elon Lindenstrauss. Symmetry of entropy in higher rank diagonalizable actions and measure classification. Journal of Modern Dynamics, 2018, 13: 163-185. doi: 10.3934/jmd.2018016

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (22)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences