# American Institute of Mathematical Sciences

December  2015, 20(10): 3415-3434. doi: 10.3934/dcdsb.2015.20.3415

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

 1 Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202, Spain

Received  December 2014 Revised  March 2015 Published  September 2015

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.
Citation: José M. Amigó, Ángel Giménez. Formulas for the topological entropy of multimodal maps based on min-max symbols. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3415-3434. doi: 10.3934/dcdsb.2015.20.3415
##### References:
 [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. [2] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific, Singapore, 2000. doi: 10.1142/4205. [3] 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. [4] 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. [5] S. L. Baldwin and E. E. Slaminka, Calculating topological entropy, J. Statist. Phys., 89 (1997), 1017-1033. doi: 10.1007/BF02764219. [6] 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. [7] 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. [8] 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. [9] 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. [10] R. Dilão, Maps of the interval, Symbolic Dynamics, Topological Entropy and Periodic Behavior (in Portuguese), Ph.D. Thesis, Instituto Superior Técnico, Lisbon, 1985. [11] 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. [12] 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. [13] 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. [14] 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. [15] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer, New York, 1993. doi: 10.1007/978-3-642-78043-1. [16] 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. [17] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63. [18] 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. [19] M. Tsujii, A simple proof for monotonicity of entropy in the quadratic family, Erg. & Dyn. Syst., 20 (2000), 925-933. doi: 10.1017/S014338570000050X. [20] P. Walters, An Introduction to Ergodic Theory, Springer Verlag, New York, 2000.

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##### References:
 [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. [2] L. Alsedà, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, World Scientific, Singapore, 2000. doi: 10.1142/4205. [3] 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. [4] 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. [5] S. L. Baldwin and E. E. Slaminka, Calculating topological entropy, J. Statist. Phys., 89 (1997), 1017-1033. doi: 10.1007/BF02764219. [6] 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. [7] 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. [8] 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. [9] 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. [10] R. Dilão, Maps of the interval, Symbolic Dynamics, Topological Entropy and Periodic Behavior (in Portuguese), Ph.D. Thesis, Instituto Superior Técnico, Lisbon, 1985. [11] 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. [12] 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. [13] 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. [14] 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. [15] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer, New York, 1993. doi: 10.1007/978-3-642-78043-1. [16] 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. [17] M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Studia Math., 67 (1980), 45-63. [18] 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. [19] M. Tsujii, A simple proof for monotonicity of entropy in the quadratic family, Erg. & Dyn. Syst., 20 (2000), 925-933. doi: 10.1017/S014338570000050X. [20] P. Walters, An Introduction to Ergodic Theory, Springer Verlag, New York, 2000.
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