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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

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
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show all references

References:
[1]

in Progress in nonlinear science, Vol. 1 (Nizhny Novgorod, 2001), RAS, Inst. Appl. Phys., Nizhniĭ Novgorod, 2002, 9-30.  Google Scholar

[2]

AMS Studies in Advance Mathematics, 28, American Mathematical Society, Providence, RI, 2003.  Google Scholar

[3]

Nonlinearity, 17 (2004), 105-116. doi: 10.1088/0951-7715/17/1/007.  Google Scholar

[4]

Chaos, 13 (2003), 519-532. doi: 10.1063/1.1566171.  Google Scholar

[5]

Phys. Rep., 75 (1981), 287-325. doi: 10.1016/0370-1573(81)90186-1.  Google Scholar

[6]

Second Edition, {World Scientific}, 2000. doi: 10.1142/4205.  Google Scholar

[7]

Erg. Th. Dyn. Syst., 4 (1984), 493-498. doi: 10.1017/S0143385700002595.  Google Scholar

[8]

Int. Journal of bifurcation and Chaos, 18 (2008), 161-168. doi: 10.1142/S0218127408020203.  Google Scholar

[9]

Nonlinearity, 16 (2003), 1214-1238. doi: 10.1088/0951-7715/16/4/302.  Google Scholar

[10]

AMS Chelsea publishing, 1959.  Google Scholar

[11]

Trans. of AMS, 351 (1999), 2927-2948. doi: 10.1090/S0002-9947-99-02344-2.  Google Scholar

[12]

Ergod. Th. & Dynam. Sys, 29 (2009), 1163-1183. doi: 10.1017/S0143385708080620.  Google Scholar

[13]

Math. Proc. Camb. Phil. Soc., 89 (1981), 107-111. doi: 10.1017/S0305004100057984.  Google Scholar

[14]

Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[15]

Usp. Mat. Nauk, 14 (1959), 3-86.  Google Scholar

[16]

PhD Thesis, SUNY et Stony Brook, 1995.  Google Scholar

[17]

in Disordered Systems and Biological Organization (Les Houches, 1985), NATO Adv. Sci. Inst. Ser. F Comput. Systems Sci., 20, Springer, Berlin, 1986, 113-115.  Google Scholar

[18]

Complex Syst., 2 (1988), 357-385.  Google Scholar

[19]

Inst. Hantes Études Sci Publ. Math., 57 (1983), 5-71.  Google Scholar

[20]

J. Combin. Theory Ser. A, 97 (2002), 129-161. doi: 10.1006/jcta.2001.3201.  Google Scholar

[21]

Combin. Probab. Comput., 13 (2004), 735-761. doi: 10.1017/S0963548304006248.  Google Scholar

[22]

SIAM Rev., 50 (2008), 199-272. doi: 10.1137/050643866.  Google Scholar

[23]

Fundamenta Mathematicae, 146 (1995), 189-201.  Google Scholar

[24]

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

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