American Institute of Mathematical Sciences

Advanced
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
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.  Google Scholar

[2]

V. Afraimovich and S. B. Hsu, Lectures on Chaotic Dynamical Systems, AMS Studies in Advance Mathematics, 28, American Mathematical Society, Providence, RI, 2003.  Google Scholar

[3]

V. Afraimovich, A. Morante and E. Ugalde, On the density of directional entropy in lattice dynamical systems, Nonlinearity, 17 (2004), 105-116. 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), 519-532. 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), 287-325. 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, {World Scientific}, 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), 493-498. 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), 161-168. doi: 10.1142/S0218127408020203.  Google Scholar

[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.  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), 2927-2948. 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), 1163-1183. doi: 10.1017/S0143385708080620.  Google Scholar

[13]

R. Ito, Rotation sets are closed, Math. Proc. Camb. Phil. Soc., 89 (1981), 107-111. doi: 10.1017/S0305004100057984.  Google Scholar

[14]

A. Katok and B. Hasselblatt, Introduction to the Morden Theory of Dynamical Systems, Cambridge University Press, Cambridge, 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), 3-86.  Google Scholar

[16]

J. Kwapisz, Rotation Sets and Entropy, PhD Thesis, SUNY et Stony Brook, 1995.  Google Scholar

[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.  Google Scholar

[18]

J. Milnor, On the entropy geometry of cellular automata, Complex Syst., 2 (1988), 357-385.  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), 5-71.  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), 129-161. 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), 735-761. 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), 199-272. doi: 10.1137/050643866.  Google Scholar

[23]

K. Ziemian, Rotation sets for subshifts of finite type, Fundamenta Mathematicae, 146 (1995), 189-201.  Google Scholar

[24]

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

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.  Google Scholar

[2]

V. Afraimovich and S. B. Hsu, Lectures on Chaotic Dynamical Systems, AMS Studies in Advance Mathematics, 28, American Mathematical Society, Providence, RI, 2003.  Google Scholar

[3]

V. Afraimovich, A. Morante and E. Ugalde, On the density of directional entropy in lattice dynamical systems, Nonlinearity, 17 (2004), 105-116. 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), 519-532. 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), 287-325. 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, {World Scientific}, 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), 493-498. 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), 161-168. doi: 10.1142/S0218127408020203.  Google Scholar

[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.  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), 2927-2948. 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), 1163-1183. doi: 10.1017/S0143385708080620.  Google Scholar

[13]

R. Ito, Rotation sets are closed, Math. Proc. Camb. Phil. Soc., 89 (1981), 107-111. doi: 10.1017/S0305004100057984.  Google Scholar

[14]

A. Katok and B. Hasselblatt, Introduction to the Morden Theory of Dynamical Systems, Cambridge University Press, Cambridge, 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), 3-86.  Google Scholar

[16]

J. Kwapisz, Rotation Sets and Entropy, PhD Thesis, SUNY et Stony Brook, 1995.  Google Scholar

[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.  Google Scholar

[18]

J. Milnor, On the entropy geometry of cellular automata, Complex Syst., 2 (1988), 357-385.  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), 5-71.  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), 129-161. 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), 735-761. 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), 199-272. doi: 10.1137/050643866.  Google Scholar

[23]

K. Ziemian, Rotation sets for subshifts of finite type, Fundamenta Mathematicae, 146 (1995), 189-201.  Google Scholar

[24]

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

[1]

Richard Miles, Thomas Ward. Directional uniformities, periodic points, and entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3525-3545. doi: 10.3934/dcdsb.2015.20.3525
[2]

Marcelo E. de Oliveira, Luiz M. G. Neto. Directional entropy based model for diffusivity-driven tumor growth. Mathematical Biosciences & Engineering, 2016, 13 (2) : 333-341. doi: 10.3934/mbe.2015005
[3]

Andreas Hiltebrand, Siddhartha Mishra. Entropy stability and well-balancedness of space-time DG for the shallow water equations with bottom topography. Networks & Heterogeneous Media, 2016, 11 (1) : 145-162. doi: 10.3934/nhm.2016.11.145
[4]

Yuming Zhang. On continuity equations in space-time domains. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4837-4873. doi: 10.3934/dcds.2018212
[5]

Rafael Monteiro. Horizontal patterns from finite speed directional quenching. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3503-3534. doi: 10.3934/dcdsb.2018285
[6]

Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181
[7]

Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 75-90. doi: 10.3934/dcds.2018004
[8]

Rasmus Dalgas Kongskov, Yiqiu Dong. Tomographic reconstruction methods for decomposing directional components. Inverse Problems & Imaging, 2018, 12 (6) : 1429-1442. doi: 10.3934/ipi.2018060
[9]

Xuemei Chen, Julia Dobrosotskaya. Inpainting via sparse recovery with directional constraints. Mathematical Foundations of Computing, 2020, 3 (4) : 229-247. doi: 10.3934/mfc.2020025
[10]

Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021, 11 (3) : 658-679. doi: 10.3934/mcrf.2021017
[11]

Vincent Astier, Thomas Unger. Galois extensions, positive involutions and an application to unitary space-time coding. Advances in Mathematics of Communications, 2019, 13 (3) : 513-516. doi: 10.3934/amc.2019032
[12]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[13]

Susanne Pumplün, Thomas Unger. Space-time block codes from nonassociative division algebras. Advances in Mathematics of Communications, 2011, 5 (3) : 449-471. doi: 10.3934/amc.2011.5.449
[14]

Gerard A. Maugin, Martine Rousseau. Prolegomena to studies on dynamic materials and their space-time homogenization. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1599-1608. doi: 10.3934/dcdss.2013.6.1599
[15]

Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713
[16]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[17]

Frédérique Oggier, B. A. Sethuraman. Quotients of orders in cyclic algebras and space-time codes. Advances in Mathematics of Communications, 2013, 7 (4) : 441-461. doi: 10.3934/amc.2013.7.441
[18]

Grégory Berhuy. Algebraic space-time codes based on division algebras with a unitary involution. Advances in Mathematics of Communications, 2014, 8 (2) : 167-189. doi: 10.3934/amc.2014.8.167
[19]

David Grant, Mahesh K. Varanasi. Duality theory for space-time codes over finite fields. Advances in Mathematics of Communications, 2008, 2 (1) : 35-54. doi: 10.3934/amc.2008.2.35
[20]

Montgomery Taylor. The diffusion phenomenon for damped wave equations with space-time dependent coefficients. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5921-5941. doi: 10.3934/dcds.2018257

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (94)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences