Advanced Search
Article Contents
Article Contents

Formulas for the topological entropy of multimodal maps based on min-max symbols

Abstract Related Papers Cited by
  • In this paper, a new formula for the topological entropy of a multimodal map $f$ is derived, and some basic properties are studied. By a formula we mean an analytical expression leading to a numerical algorithm; by a multimodal map we mean a continuous interval self-map which is strictly monotonic in a finite number of subintervals. The main feature of this formula is that it involves the min-max symbols of $f$, which are closely related to its kneading symbols. This way we continue our pursuit of finding expressions for the topological entropy of continuous multimodal maps based on min-max symbols. As in previous cases, which will be also reviewed, the main geometrical ingredients of the new formula are the numbers of transversal crossings of the graph of $f$ and its iterates with the so-called "critical lines". The theoretical and practical underpinnings are worked out with the family of logistic parabolas and numerical simulations.
    Mathematics Subject Classification: Primary: 37B40, 37E05; Secondary: 37B10, 65Dxx.


    \begin{equation} \\ \end{equation}
  • [1]

    R. Adler, A. Konheim and M. McAndrew, Topological entropy, Trans. Amer. Mat. Soc., 114 (1965), 309-319.doi: 10.1090/S0002-9947-1965-0175106-9.


    L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific, Singapore, 2000.doi: 10.1142/4205.


    J. M. Amigó, R. Dilão and A. Giménez, Computing the topological entropy of multimodal maps via Min-Max sequences, Entropy, 14 (2012), 742-768.doi: 10.3390/e14040742.


    J. M. Amigó and A. Giménez, A Simplified algorithm for the topological entropy of multimodal maps, Entropy, 16 (2014), 627-644.doi: 10.3390/e16020627.


    S. L. Baldwin and E. E. Slaminka, Calculating topological entropy, J. Statist. Phys., 89 (1997), 1017-1033.doi: 10.1007/BF02764219.


    L. Block, J. Keesling, S. Li and K. Peterson, An improved algorithm for computing topological entropy, J. Statist. Phys., 55 (1989), 929-939.doi: 10.1007/BF01041072.


    L. Block and J. Keesling, Computing the topological entropy of maps pf the interval with three monotone pieces, J. Statist. Phys., 66 (1991), 755-774.doi: 10.1007/BF01055699.


    P. Collet, J. P. Crutchfield and J. P. Eckmann, Computing the topological entropy of maps, Comm. Math. Phys., 88 (1983), 257-262.doi: 10.1007/BF01209479.


    J. Dias de Deus, R. Dilão and J. Taborda Duarte, Topological entropy and approaches to chaos in dynamics of the interval, Phys. Lett., 90 (1982), 1-4.doi: 10.1016/0375-9601(82)90033-0.


    R. Dilão, Maps of the interval, Symbolic Dynamics, Topological Entropy and Periodic Behavior (in Portuguese), Ph.D. Thesis, Instituto Superior Técnico, Lisbon, 1985.


    R. Dilão and J. M. Amigó, Computing the topological entropy of unimodal maps, International Journal of Bifurcations and Chaos, 22 (2012), 1250152, 14pp.doi: 10.1142/S0218127412501520.


    A. Douady, Topological entropy of unimodal maps: Monotonicity for cuadratic polynomials, in Real and Complex Dynamical Systems (eds. B. Branner and P. Hjorth), 464, Kluwer, 1995, 65-87.


    G. Froyland, R. Murray and D. Terhesiu, Efficient computation of topological entropy, pressure, conformal measures, and equilibrium states in one dimension, Phys. Rev. E, 76 (2007), 036702, 5pp.doi: 10.1103/PhysRevE.76.036702.


    P. Góra and A. Boyarsky, Computing the topological entropy of general one-dimensional maps, Trans. Amer. Math. Soc., 323 (1991), 39-49.doi: 10.1090/S0002-9947-1991-1062871-7.


    W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer, New York, 1993.doi: 10.1007/978-3-642-78043-1.


    J. Milnor and W. Thurston, On iterated maps of the interval, in Dynamical Systems (ed. J. C. Alexander), Lectures Notes in Mathematics, 1342, Springer, 1988, 465-563.doi: 10.1007/BFb0082847.


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


    T. Steinberger, Computing the topological entropy for piecewise monotonic maps on the interval, J. Statist. Phys., 95 (1999), 287-303.doi: 10.1023/A:1004585613252.


    M. Tsujii, A simple proof for monotonicity of entropy in the quadratic family, Erg. & Dyn. Syst., 20 (2000), 925-933.doi: 10.1017/S014338570000050X.


    P. Walters, An Introduction to Ergodic Theory, Springer Verlag, New York, 2000.

  • 加载中

Article Metrics

HTML views() PDF downloads(274) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint