Directional complexity and entropy for lift mappings

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December 2015, 20(10): 3385-3401. doi: 10.3934/dcdsb.2015.20.3385

Directional complexity and entropy for lift mappings

Valentin Afraimovich ^1, , Maurice Courbage ^2, and Lev Glebsky ^3,

1.	Instituto de Investigación en Comunicación Óptica, Universidad Autónoma de San Luis Potosí, Karakorum 1470, Lomas 4a 78220, San Luis Potosi, S.L.P, Mexico
2.	Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057 CNRS et Université Paris 7-Denis Diderot, 10, rue Alice Domon et Léonie Duquet 75205 Paris Cedex 13, France
3.	Instituto de Investigación en Comunicación Óptica, Universidad Autónoma de San Luis Potos, Karakorum 1470, Lomas 4a 78220, San Luis Potosi, S.L.P, Mexico

Received December 2014 Revised March 2015 Published September 2015

Abstract
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We introduce and study the notion of a directional complexity and entropy for maps of degree $1$ on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a combinatorial machinery to obtain exact formulas for the directional entropy, to find the maximal directional entropy, and to show that it equals the topological entropy of the map.

Keywords: directional entropy., space-time window, Rotation interval, directional complexity.

Mathematics Subject Classification: 37E10, 37E4.

Citation: Valentin Afraimovich, Maurice Courbage, Lev Glebsky. Directional complexity and entropy for lift mappings. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3385-3401. doi: 10.3934/dcdsb.2015.20.3385

References:

[1]	V. Afraimovich, M. Courbage, B. Fernandez and A. Morante, Directional entropy in lattice dynamical systems,, in Progress in nonlinear science, (2001), 9. Google Scholar
[2]	V. Afraimovich and S. B. Hsu, Lectures on Chaotic Dynamical Systems,, AMS Studies in Advance Mathematics, (2003). Google Scholar
[3]	V. Afraimovich, A. Morante and E. Ugalde, On the density of directional entropy in lattice dynamical systems,, Nonlinearity, 17 (2004), 105. doi: 10.1088/0951-7715/17/1/007. Google Scholar
[4]	V. Afraimovich and G. M. Zaslavsky, Space-time complexity in Hamiltonian dynamics,, Chaos, 13 (2003), 519. doi: 10.1063/1.1566171. Google Scholar
[5]	V. M. Alekseev and M. V. Yakobson, Symbolic dynamics and hyperbolic dynamic systems,, Phys. Rep., 75 (1981), 287. doi: 10.1016/0370-1573(81)90186-1. Google Scholar
[6]	L. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, Second Edition, (2000). doi: 10.1142/4205. Google Scholar
[7]	R. Bamon, I. P. Malta, M. J. Pacifico and F. Takens, Rotation intervals of endomorphisms of the circle,, Erg. Th. Dyn. Syst., 4 (1984), 493. doi: 10.1017/S0143385700002595. Google Scholar
[8]	M. Courbage and B. Kaminski, Density of measure-theoretical directional entropy for lattice dynamical systems,, Int. Journal of bifurcation and Chaos, 18 (2008), 161. doi: 10.1142/S0218127408020203. Google Scholar
[9]	S. Galatolo, Complexity, initial condition sensitivity, dimensions and weak chaos in dynamical systems,, Nonlinearity, 16 (2003), 1214. doi: 10.1088/0951-7715/16/4/302. Google Scholar
[10]	F. Gantmacher, Theory of Matrices,, AMS Chelsea publishing, (1959). Google Scholar
[11]	W. Geller and M. Misiurewicz, Rotation and entropy,, Trans. of AMS, 351 (1999), 2927. doi: 10.1090/S0002-9947-99-02344-2. Google Scholar
[12]	E. Gutkin and M. Rams, Growth rates for geometric complexities and counting functions in polygonal billiards,, Ergod. Th. & Dynam. Sys, 29 (2009), 1163. doi: 10.1017/S0143385708080620. Google Scholar
[13]	R. Ito, Rotation sets are closed,, Math. Proc. Camb. Phil. Soc., 89 (1981), 107. doi: 10.1017/S0305004100057984. Google Scholar
[14]	A. Katok and B. Hasselblatt, Introduction to the Morden Theory of Dynamical Systems,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511809187. Google Scholar
[15]	A. N. Kolmogorov and V. M. Tikhomirov, $\epsilon$-entropy and $\epsilon$-capacity of sets in functional spaces,, Usp. Mat. Nauk, 14 (1959), 3. Google Scholar
[16]	J. Kwapisz, Rotation Sets and Entropy,, PhD Thesis, (1995). Google Scholar
[17]	J. Milnor, Directional entropy of cellular automaton maps,, in Disordered Systems and Biological Organization* (Les Houches*, (1985), 113. Google Scholar
[18]	J. Milnor, On the entropy geometry of cellular automata,, Complex Syst., 2 (1988), 357. Google Scholar
[19]	S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms,, Inst. Hantes Études Sci Publ. Math., 57 (1983), 5. Google Scholar
[20]	R. Pemantle and M. Wilson, Asymptotics of multivariate sequences, I. Smooth points of the singular variety,, J. Combin. Theory Ser. A, 97 (2002), 129. doi: 10.1006/jcta.2001.3201. Google Scholar
[21]	R. Pemantle and M. Wilson, Asymptotics of multivariate sequences, II. Multiple points of the singular variety,, Combin. Probab. Comput., 13 (2004), 735. doi: 10.1017/S0963548304006248. Google Scholar
[22]	R. Pemantle and M. Wilson, Twenty combinatorial esxamples of asymptotics derived from multivariate generating functions,, SIAM Rev., 50 (2008), 199. doi: 10.1137/050643866. Google Scholar
[23]	K. Ziemian, Rotation sets for subshifts of finite type,, Fundamenta Mathematicae, 146 (1995), 189. Google Scholar
[24]	, SAGE is an open source mathematics software,, See , (). Google Scholar

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References:

[1]	V. Afraimovich, M. Courbage, B. Fernandez and A. Morante, Directional entropy in lattice dynamical systems,, in Progress in nonlinear science, (2001), 9. Google Scholar
[2]	V. Afraimovich and S. B. Hsu, Lectures on Chaotic Dynamical Systems,, AMS Studies in Advance Mathematics, (2003). Google Scholar
[3]	V. Afraimovich, A. Morante and E. Ugalde, On the density of directional entropy in lattice dynamical systems,, Nonlinearity, 17 (2004), 105. doi: 10.1088/0951-7715/17/1/007. Google Scholar
[4]	V. Afraimovich and G. M. Zaslavsky, Space-time complexity in Hamiltonian dynamics,, Chaos, 13 (2003), 519. doi: 10.1063/1.1566171. Google Scholar
[5]	V. M. Alekseev and M. V. Yakobson, Symbolic dynamics and hyperbolic dynamic systems,, Phys. Rep., 75 (1981), 287. doi: 10.1016/0370-1573(81)90186-1. Google Scholar
[6]	L. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, Second Edition, (2000). doi: 10.1142/4205. Google Scholar
[7]	R. Bamon, I. P. Malta, M. J. Pacifico and F. Takens, Rotation intervals of endomorphisms of the circle,, Erg. Th. Dyn. Syst., 4 (1984), 493. doi: 10.1017/S0143385700002595. Google Scholar
[8]	M. Courbage and B. Kaminski, Density of measure-theoretical directional entropy for lattice dynamical systems,, Int. Journal of bifurcation and Chaos, 18 (2008), 161. doi: 10.1142/S0218127408020203. Google Scholar
[9]	S. Galatolo, Complexity, initial condition sensitivity, dimensions and weak chaos in dynamical systems,, Nonlinearity, 16 (2003), 1214. doi: 10.1088/0951-7715/16/4/302. Google Scholar
[10]	F. Gantmacher, Theory of Matrices,, AMS Chelsea publishing, (1959). Google Scholar
[11]	W. Geller and M. Misiurewicz, Rotation and entropy,, Trans. of AMS, 351 (1999), 2927. doi: 10.1090/S0002-9947-99-02344-2. Google Scholar
[12]	E. Gutkin and M. Rams, Growth rates for geometric complexities and counting functions in polygonal billiards,, Ergod. Th. & Dynam. Sys, 29 (2009), 1163. doi: 10.1017/S0143385708080620. Google Scholar
[13]	R. Ito, Rotation sets are closed,, Math. Proc. Camb. Phil. Soc., 89 (1981), 107. doi: 10.1017/S0305004100057984. Google Scholar
[14]	A. Katok and B. Hasselblatt, Introduction to the Morden Theory of Dynamical Systems,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511809187. Google Scholar
[15]	A. N. Kolmogorov and V. M. Tikhomirov, $\epsilon$-entropy and $\epsilon$-capacity of sets in functional spaces,, Usp. Mat. Nauk, 14 (1959), 3. Google Scholar
[16]	J. Kwapisz, Rotation Sets and Entropy,, PhD Thesis, (1995). Google Scholar
[17]	J. Milnor, Directional entropy of cellular automaton maps,, in Disordered Systems and Biological Organization* (Les Houches*, (1985), 113. Google Scholar
[18]	J. Milnor, On the entropy geometry of cellular automata,, Complex Syst., 2 (1988), 357. Google Scholar
[19]	S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms,, Inst. Hantes Études Sci Publ. Math., 57 (1983), 5. Google Scholar
[20]	R. Pemantle and M. Wilson, Asymptotics of multivariate sequences, I. Smooth points of the singular variety,, J. Combin. Theory Ser. A, 97 (2002), 129. doi: 10.1006/jcta.2001.3201. Google Scholar
[21]	R. Pemantle and M. Wilson, Asymptotics of multivariate sequences, II. Multiple points of the singular variety,, Combin. Probab. Comput., 13 (2004), 735. doi: 10.1017/S0963548304006248. Google Scholar
[22]	R. Pemantle and M. Wilson, Twenty combinatorial esxamples of asymptotics derived from multivariate generating functions,, SIAM Rev., 50 (2008), 199. doi: 10.1137/050643866. Google Scholar
[23]	K. Ziemian, Rotation sets for subshifts of finite type,, Fundamenta Mathematicae, 146 (1995), 189. Google Scholar
[24]	, SAGE is an open source mathematics software,, See , (). Google Scholar

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