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

Directional complexity and entropy for lift mappings

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

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.
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.

[2]

V. Afraimovich and S. B. Hsu, Lectures on Chaotic Dynamical Systems,, AMS Studies in Advance Mathematics, (2003).

[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.

[4]

V. Afraimovich and G. M. Zaslavsky, Space-time complexity in Hamiltonian dynamics,, Chaos, 13 (2003), 519. doi: 10.1063/1.1566171.

[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.

[6]

L. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, Second Edition, (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. 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. doi: 10.1142/S0218127408020203.

[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.

[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. 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. doi: 10.1017/S0143385708080620.

[13]

R. Ito, Rotation sets are closed,, Math. Proc. Camb. Phil. Soc., 89 (1981), 107. doi: 10.1017/S0305004100057984.

[14]

A. Katok and B. Hasselblatt, Introduction to the Morden Theory of Dynamical Systems,, Cambridge University Press, (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.

[16]

J. Kwapisz, Rotation Sets and Entropy,, PhD Thesis, (1995).

[17]

J. Milnor, Directional entropy of cellular automaton maps,, in Disordered Systems and Biological Organization (Les Houches, (1985), 113.

[18]

J. Milnor, On the entropy geometry of cellular automata,, Complex Syst., 2 (1988), 357.

[19]

S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms,, Inst. Hantes Études Sci Publ. Math., 57 (1983), 5.

[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.

[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.

[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.

[23]

K. Ziemian, Rotation sets for subshifts of finite type,, Fundamenta Mathematicae, 146 (1995), 189.

[24]

, SAGE is an open source mathematics software,, See , ().

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, (2001), 9.

[2]

V. Afraimovich and S. B. Hsu, Lectures on Chaotic Dynamical Systems,, AMS Studies in Advance Mathematics, (2003).

[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.

[4]

V. Afraimovich and G. M. Zaslavsky, Space-time complexity in Hamiltonian dynamics,, Chaos, 13 (2003), 519. doi: 10.1063/1.1566171.

[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.

[6]

L. Alseda, J. Llibre and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One,, Second Edition, (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. 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. doi: 10.1142/S0218127408020203.

[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.

[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. 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. doi: 10.1017/S0143385708080620.

[13]

R. Ito, Rotation sets are closed,, Math. Proc. Camb. Phil. Soc., 89 (1981), 107. doi: 10.1017/S0305004100057984.

[14]

A. Katok and B. Hasselblatt, Introduction to the Morden Theory of Dynamical Systems,, Cambridge University Press, (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.

[16]

J. Kwapisz, Rotation Sets and Entropy,, PhD Thesis, (1995).

[17]

J. Milnor, Directional entropy of cellular automaton maps,, in Disordered Systems and Biological Organization (Les Houches, (1985), 113.

[18]

J. Milnor, On the entropy geometry of cellular automata,, Complex Syst., 2 (1988), 357.

[19]

S. Newhouse, J. Palis and F. Takens, Bifurcations and stability of families of diffeomorphisms,, Inst. Hantes Études Sci Publ. Math., 57 (1983), 5.

[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.

[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.

[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.

[23]

K. Ziemian, Rotation sets for subshifts of finite type,, Fundamenta Mathematicae, 146 (1995), 189.

[24]

, SAGE is an open source mathematics software,, See , ().

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