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 & 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.  doi: 10.1090/S0002-9947-1965-0175106-9.  Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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.  doi: 10.1007/BF01055699.  Google Scholar

[8]

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

[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.  doi: 10.1016/0375-9601(82)90033-0.  Google Scholar

[10]

R. Dilão, Maps of the interval, Symbolic Dynamics, Topological Entropy and Periodic Behavior (in Portuguese),, Ph.D. Thesis, (1985).   Google Scholar

[11]

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

[12]

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

[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).  doi: 10.1103/PhysRevE.76.036702.  Google Scholar

[14]

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

[15]

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

[16]

J. Milnor and W. Thurston, On iterated maps of the interval,, in Dynamical Systems (ed. J. C. Alexander), (1342), 465.  doi: 10.1007/BFb0082847.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

P. Walters, An Introduction to Ergodic Theory,, Springer Verlag, (2000).   Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[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.  doi: 10.1007/BF01055699.  Google Scholar

[8]

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

[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.  doi: 10.1016/0375-9601(82)90033-0.  Google Scholar

[10]

R. Dilão, Maps of the interval, Symbolic Dynamics, Topological Entropy and Periodic Behavior (in Portuguese),, Ph.D. Thesis, (1985).   Google Scholar

[11]

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

[12]

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

[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).  doi: 10.1103/PhysRevE.76.036702.  Google Scholar

[14]

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

[15]

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

[16]

J. Milnor and W. Thurston, On iterated maps of the interval,, in Dynamical Systems (ed. J. C. Alexander), (1342), 465.  doi: 10.1007/BFb0082847.  Google Scholar

[17]

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

[18]

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

[19]

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

[20]

P. Walters, An Introduction to Ergodic Theory,, Springer Verlag, (2000).   Google Scholar

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