Indecomposable Coupled Map Lattices with Non–unique Phase

  • R.S. MacKay
Part of the Lecture Notes in Physics book series (LNP, volume 671)


A compact topologically mixing uniformly hyperbolic attractor for a diffeomorphism of a finite-dimensional manifold with Hölder continuous derivative carries a unique probability measure describing the long-time statistics of the forward orbits of almost all initial conditions in its basin. This was proved [28] by converting the orbits of the dynamical system into symbol sequences from a finite alphabet (indexed by time) and considering them as states of a statistical mechanical spin chain with a certain interaction energy which decays exponentially with separation, so leading to a unique phase. The results have been extended to many classes of non-uniformly hyperbolic system by different approaches (e.g. [30]).


Markov Chain Symbolic Dynamic Symbol Sequence Symbolic State Unique Phase 
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  1. 1.
    J.-B. Bardet, Limit theorems for coupled analytic maps, Probab. Th. Rel. Fields 124 (2002) 151–177.CrossRefGoogle Scholar
  2. 2.
    C. Beck, Spatio-temporal chaos and vacuum fluctuations of quantized fields (World Sci, 2002).Google Scholar
  3. 3.
    C. Bennett, G. Grinstein, Role of irreversibility in stabilizing complex and nonergodic behaviour in locally interacting discrete systems, Phys. Rev. Lett. 55 (1985) 657–660.CrossRefPubMedGoogle Scholar
  4. 4.
    J. Bricmont, A. Kupiainen, High temperature expansions and dynamical systems, Commun. Math. Phys. 178 (1996) 703–32.Google Scholar
  5. 5.
    L. A. Bunimovich, Ya. G. Sinai, Space-time chaos in coupled map lattices, Nonlinearity 1 (1988) 491–516.CrossRefGoogle Scholar
  6. 6.
    R. Burton R. J. E. Steif, Non-uniqueness of measures of maximal entropy for subshifts of finite type, Erg. Th. Dyn. Sys. 14 (1994) 213–35.Google Scholar
  7. 7.
    H. Chaté, P. Manneville, Collective behaviors in spatially extended systems with local interactions and synchronous updating, Prog. Theor. Phys. 87 (1992) 1–60.Google Scholar
  8. 8.
    J. P. Crutchfield, K. Kaneko, Are attractors relevant to turbulence?, Phys. Rev. Lett. 60 (1988) 2715–8.CrossRefPubMedGoogle Scholar
  9. 9.
    P. Gacs, Reliable cellular automata with self-organization, J. Stat. Phys. 103 (2001) 45–267.CrossRefGoogle Scholar
  10. 10.
    G. Gielis, R. S. MacKay, Coupled map lattices with phase transition, Nonlinearity 13 (2000) 867–888.CrossRefGoogle Scholar
  11. 11.
    F. Ginelli, R. Livi, A. Politi, Emergence of chaotic behaviour in linearly stable systems, J. Phys. A 35 (2002) 499–516.CrossRefMathSciNetGoogle Scholar
  12. 12.
    L. F. Gray, A reader's guide to Gacs' “positive rates'' paper, J. Stat. Phys. 103 (2001) 1–44.CrossRefGoogle Scholar
  13. 13.
    H. Haken, Cooperative phenomena in systems far from thermal equilibrium and in nonphysical systems, Rev. Mod. Phys. 47 (1975) 67–121.CrossRefGoogle Scholar
  14. 14.
    M. Jiang, Ya. B. Pesin, Equilibrium measures for coupled map lattices: existence, uniqueness and finite dimensional approximations, Commun. Math. Phys. 193 (1998) 677–711.CrossRefGoogle Scholar
  15. 15.
    K. Kaneko, Period-doubling of kink-antikink patterns, quasiperiodicity in antiferro-like structures and spatial intermittency in coupled map lattices, Prog. Theor. Phys. 72 (1984) 480–486.Google Scholar
  16. 16.
    K. Kaneko, Globally coupled chaos violates the law of large numbers, Phys. Rev. Lett. 65 (1990) 1391–4.CrossRefPubMedGoogle Scholar
  17. 17.
    A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems (Cambridge Univ Press, 1995).Google Scholar
  18. 18.
    F. P. Kelly, Reversibility and stochastic networks (Wiley, 1979).Google Scholar
  19. 19.
    Y. Kuramoto, Cooperative dynamics of oscillator community, Prog. Theor. Phys. 79 (1984) 223–240.Google Scholar
  20. 20.
    J. L. Lebowitz, C. Maes, E. Speer, Statistical mechanics of probabilistic cellular automata, J. Stat. Phys. 59 (1990) 117–70.CrossRefGoogle Scholar
  21. 21.
    T. M. Liggett, Interacting particle systems (Springer, 1985).Google Scholar
  22. 22.
    J. Losson, M. C. Mackey, Thermodynamic properties of coupled map lattices, in: Stochastic and spatial structures of dynamical systems, eds. S. J. van Strien, S. M. Verduyn Lunel (North Holland, 1996), 41–69.Google Scholar
  23. 23.
    R. S. MacKay, Dynamics of networks: features that persist from the uncoupled limit, in “Stochastic and spatial structures of dynamical systems'', eds. S. J. van Strien, S. M. Verduyn Lunel (North Holland, 1996), 81–104.Google Scholar
  24. 24.
    J. Miller, D. A. Huse, Macroscopic equilibrium from microscopic irreversibility in a chaotic coupled map lattice, Phys. Rev. E 48 (1993) 2528–35.CrossRefGoogle Scholar
  25. 25.
    R. Penrose, The road to reality (Random House, 2004)Google Scholar
  26. 26.
    Ya. B. Pesin, Ya. G. Sinai, Space-time chaos in chains of weakly coupled interacting hyperbolic mappings, Adv Sov Math 3 (1991) 165–198.Google Scholar
  27. 27.
    H. Sakaguchi, Phase transition in coupled Bernoulli maps, Prog. Theor. Phys. 80 (1988) 7–12.Google Scholar
  28. 28.
    Ya. G. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surv. 27:4 (1972) 21–69.Google Scholar
  29. 29.
    A. L. Toom, N. B. Vasilyev, O. N. Stavskaya, L. G. Mityushin, G. L. Kurdyumov, S. A. Pirogov, Discrete local Markov systems. In: Stochastic cellular systems, ergodicity, memory, morphogenesis, eds R. L. Dobrushin, V. I. Kryukov, A. L. Toom (Manchester Univ Press, 1990), 1–182.Google Scholar
  30. 30.
    M. Viana, Stochastic dynamics of deterministic systems (Instituto de Matematica Pura e Aplicada, 1997).Google Scholar

Authors and Affiliations

  • R.S. MacKay
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickU.K.

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