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 and 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, Vol. 1 (Nizhny Novgorod, 2001), RAS, Inst. Appl. Phys., Nizhniĭ Novgorod, 2002, 9-30.
[2]	V. Afraimovich and S. B. Hsu, Lectures on Chaotic Dynamical Systems, AMS Studies in Advance Mathematics, 28, American Mathematical Society, Providence, RI, 2003.
[3]	V. Afraimovich, A. Morante and E. Ugalde, On the density of directional entropy in lattice dynamical systems, Nonlinearity, 17 (2004), 105-116. doi: 10.1088/0951-7715/17/1/007.
[4]	V. Afraimovich and G. M. Zaslavsky, Space-time complexity in Hamiltonian dynamics, Chaos, 13 (2003), 519-532. doi: 10.1063/1.1566171.
[5]	V. M. Alekseev and M. V. Yakobson, Symbolic dynamics and hyperbolic dynamic systems, Phys. Rep., 75 (1981), 287-325. doi: 10.1016/0370-1573(81)90186-1.
[6]	L. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Second Edition, {World Scientific}, 2000. doi: 10.1142/4205.
[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), 493-498. doi: 10.1017/S0143385700002595.
[8]	M. Courbage and B. Kaminski, Density of measure-theoretical directional entropy for lattice dynamical systems, Int. Journal of bifurcation and Chaos, 18 (2008), 161-168. doi: 10.1142/S0218127408020203.
[9]	S. Galatolo, Complexity, initial condition sensitivity, dimensions and weak chaos in dynamical systems, Nonlinearity, 16 (2003), 1214-1238. doi: 10.1088/0951-7715/16/4/302.
[10]	F. Gantmacher, Theory of Matrices, AMS Chelsea publishing, 1959.
[11]	W. Geller and M. Misiurewicz, Rotation and entropy, Trans. of AMS, 351 (1999), 2927-2948. doi: 10.1090/S0002-9947-99-02344-2.
[12]	E. Gutkin and M. Rams, Growth rates for geometric complexities and counting functions in polygonal billiards, Ergod. Th. & Dynam. Sys, 29 (2009), 1163-1183. doi: 10.1017/S0143385708080620.
[13]	R. Ito, Rotation sets are closed, Math. Proc. Camb. Phil. Soc., 89 (1981), 107-111. doi: 10.1017/S0305004100057984.
[14]	A. Katok and B. Hasselblatt, Introduction to the Morden Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[15]	A. N. Kolmogorov and V. M. Tikhomirov, $\epsilon$-entropy and $\epsilon$-capacity of sets in functional spaces, Usp. Mat. Nauk, 14 (1959), 3-86.
[16]	J. Kwapisz, Rotation Sets and Entropy, PhD Thesis, SUNY et Stony Brook, 1995.
[17]	J. Milnor, Directional entropy of cellular automaton maps, in Disordered Systems and Biological Organization (Les Houches, 1985), NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., 20, Springer, Berlin, 1986, 113-115.
[18]	J. Milnor, On the entropy geometry of cellular automata, Complex Syst., 2 (1988), 357-385.
[19]	S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hantes Études Sci Publ. Math., 57 (1983), 5-71.
[20]	R. Pemantle and M. Wilson, Asymptotics of multivariate sequences, I. Smooth points of the singular variety, J. Combin. Theory Ser. A, 97 (2002), 129-161. doi: 10.1006/jcta.2001.3201.
[21]	R. Pemantle and M. Wilson, Asymptotics of multivariate sequences, II. Multiple points of the singular variety, Combin. Probab. Comput., 13 (2004), 735-761. doi: 10.1017/S0963548304006248.
[22]	R. Pemantle and M. Wilson, Twenty combinatorial esxamples of asymptotics derived from multivariate generating functions, SIAM Rev., 50 (2008), 199-272. doi: 10.1137/050643866.
[23]	K. Ziemian, Rotation sets for subshifts of finite type, Fundamenta Mathematicae, 146 (1995), 189-201.
[24]	, SAGE is an open source mathematics software, See http://www.sagemath.org/.

show all references

References:

[1]	V. Afraimovich, M. Courbage, B. Fernandez and A. Morante, Directional entropy in lattice dynamical systems, in Progress in nonlinear science, Vol. 1 (Nizhny Novgorod, 2001), RAS, Inst. Appl. Phys., Nizhniĭ Novgorod, 2002, 9-30.
[2]	V. Afraimovich and S. B. Hsu, Lectures on Chaotic Dynamical Systems, AMS Studies in Advance Mathematics, 28, American Mathematical Society, Providence, RI, 2003.
[3]	V. Afraimovich, A. Morante and E. Ugalde, On the density of directional entropy in lattice dynamical systems, Nonlinearity, 17 (2004), 105-116. doi: 10.1088/0951-7715/17/1/007.
[4]	V. Afraimovich and G. M. Zaslavsky, Space-time complexity in Hamiltonian dynamics, Chaos, 13 (2003), 519-532. doi: 10.1063/1.1566171.
[5]	V. M. Alekseev and M. V. Yakobson, Symbolic dynamics and hyperbolic dynamic systems, Phys. Rep., 75 (1981), 287-325. doi: 10.1016/0370-1573(81)90186-1.
[6]	L. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, Second Edition, {World Scientific}, 2000. doi: 10.1142/4205.
[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), 493-498. doi: 10.1017/S0143385700002595.
[8]	M. Courbage and B. Kaminski, Density of measure-theoretical directional entropy for lattice dynamical systems, Int. Journal of bifurcation and Chaos, 18 (2008), 161-168. doi: 10.1142/S0218127408020203.
[9]	S. Galatolo, Complexity, initial condition sensitivity, dimensions and weak chaos in dynamical systems, Nonlinearity, 16 (2003), 1214-1238. doi: 10.1088/0951-7715/16/4/302.
[10]	F. Gantmacher, Theory of Matrices, AMS Chelsea publishing, 1959.
[11]	W. Geller and M. Misiurewicz, Rotation and entropy, Trans. of AMS, 351 (1999), 2927-2948. doi: 10.1090/S0002-9947-99-02344-2.
[12]	E. Gutkin and M. Rams, Growth rates for geometric complexities and counting functions in polygonal billiards, Ergod. Th. & Dynam. Sys, 29 (2009), 1163-1183. doi: 10.1017/S0143385708080620.
[13]	R. Ito, Rotation sets are closed, Math. Proc. Camb. Phil. Soc., 89 (1981), 107-111. doi: 10.1017/S0305004100057984.
[14]	A. Katok and B. Hasselblatt, Introduction to the Morden Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[15]	A. N. Kolmogorov and V. M. Tikhomirov, $\epsilon$-entropy and $\epsilon$-capacity of sets in functional spaces, Usp. Mat. Nauk, 14 (1959), 3-86.
[16]	J. Kwapisz, Rotation Sets and Entropy, PhD Thesis, SUNY et Stony Brook, 1995.
[17]	J. Milnor, Directional entropy of cellular automaton maps, in Disordered Systems and Biological Organization (Les Houches, 1985), NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., 20, Springer, Berlin, 1986, 113-115.
[18]	J. Milnor, On the entropy geometry of cellular automata, Complex Syst., 2 (1988), 357-385.
[19]	S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms, Inst. Hantes Études Sci Publ. Math., 57 (1983), 5-71.
[20]	R. Pemantle and M. Wilson, Asymptotics of multivariate sequences, I. Smooth points of the singular variety, J. Combin. Theory Ser. A, 97 (2002), 129-161. doi: 10.1006/jcta.2001.3201.
[21]	R. Pemantle and M. Wilson, Asymptotics of multivariate sequences, II. Multiple points of the singular variety, Combin. Probab. Comput., 13 (2004), 735-761. doi: 10.1017/S0963548304006248.
[22]	R. Pemantle and M. Wilson, Twenty combinatorial esxamples of asymptotics derived from multivariate generating functions, SIAM Rev., 50 (2008), 199-272. doi: 10.1137/050643866.
[23]	K. Ziemian, Rotation sets for subshifts of finite type, Fundamenta Mathematicae, 146 (1995), 189-201.
[24]	, SAGE is an open source mathematics software, See http://www.sagemath.org/.

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