SRB-Measures for Coupled Map Lattices

  • E Järvenpää
Part of the Lecture Notes in Physics book series (LNP, volume 671)


This chapter is divided into three parts. I start with a review of the existence results of SRB–measures for coupled map lattices. In the second part I give a brief introduction to the behaviour of Hausdor. dimension under general projections. Finally, I construct a counterexample to the Bricmont–Kupiainen conjecture and discuss its role in the definition of SRB–measures for coupled map lattices.


Lebesgue Measure Invariant Measure Local Dynamic Nikodym Derivative Bernoulli Convolution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    V. Baladi: Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics16, (World Scientific Publishing Co. Inc., River Edge NJ 2000)Google Scholar
  2. 2.
    V. Baladi: Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems 18, (1998), pp 255 –292CrossRefGoogle Scholar
  3. 3.
    V. Baladi, M. Degli Esposti, S. Isola, E. Järvenpää, and A. Kupiainen: The spectrum of weakly coupled map lattices, J. Math. Pures Appl. 77, (1998), pp 539 –584CrossRefGoogle Scholar
  4. 4.
    V. Baladi and H.H. Rugh: Floquet spectrum of weakly coupled map lattices, Comm. Math. Phys. 220, (2001), pp 561 –582Google Scholar
  5. 5.
    J.-B. Bardet: Limit theorems for coupled analytic maps, Probab. Theory Related Fields 124, (2002), pp 151 –177CrossRefGoogle Scholar
  6. 6.
    M. Blank and L. Bunimovich: Multicomponent dynamical systems: SRB measures and phase transitions, Nonlinearity 16, (2003), pp 387 –401CrossRefGoogle Scholar
  7. 7.
    M. Blank, G. Keller, and C. Liverani: Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity 15, (2002), pp 1905 –1973CrossRefGoogle Scholar
  8. 8.
    R. Bowen: Equilibrium states and ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes in Math. 470, (1975).Google Scholar
  9. 9.
    J. Bricmont and A. Kupiainen: Coupled analytic maps, Nonlinearity 8, (1995), pp 379 –396CrossRefGoogle Scholar
  10. 10.
    J. Bricmont and A. Kupiainen: High temperature expansions and dynamical systems, Comm. Math. Phys. 178, (1996), pp 703 –732Google Scholar
  11. 11.
    L.A. Bunimovich: Coupled map lattices: one step forward and two steps back, Chaos, order and patterns: aspects of nonlinearity –the “gran finale" (Como, 1993), Phys. D 86, (1995), pp 248 –255Google Scholar
  12. 12.
    L.A. Bunimovich and Ya.G. Sinai: Spacetime chaos in coupled map lattices, Nonlinearity 1, (1988), pp 491 –516CrossRefGoogle Scholar
  13. 13.
    P. Collet: Some ergodic properties of maps of the interval, Dynamical systems (Temuco, 1991/1992), Travaux en Cours 52 (Hermann Paris 1996) pp 55 –91Google Scholar
  14. 14.
    R.L. Dobrushin: Gibbsian random fields for lattice systems with pairwise interactions, Funct. Anal. Appl. 2, (1968), pp 292 –301CrossRefGoogle Scholar
  15. 15.
    R.L. Dobrushin: The description of a random field by means of conditional probabilities and conditions on its regularity. Theory Prob. Appl. 13, (1968), pp 197 –224CrossRefGoogle Scholar
  16. 16.
    K.J. Falconer: Hausdorff dimension and the exceptional set of projections, Mathematika 29, (1982), pp 109 –115Google Scholar
  17. 17.
    K.J. Falconer: Geometry of Fractal Sets, (Cambridge University Press, Cambridge 1985)Google Scholar
  18. 18.
    K.J. Falconer and J.D. Howroyd: Projection theorems for box and packing dimensions, Math. Proc. Cambridge Philos. Soc. 119, (1996), pp 287 –295Google Scholar
  19. 19.
    K.J. Falconer and J.D. Howroyd: Packing dimensions of projections and dimension profiles, Math. Proc. Cambridge Philos. Soc. 121, (1997), pp 269 –286CrossRefGoogle Scholar
  20. 20.
    K.J. Falconer and P. Mattila: The packing dimension of projections and sections of measures, Math. Proc. Cambridge Philos. Soc. 119, (1996), pp 695 –713Google Scholar
  21. 21.
    K.J. Falconer and T.C. O'Neil: Convolutions and the geometry of multifractal measures, Math. Nachr. 204, (1999), pp 61 –82Google Scholar
  22. 22.
    T. Fischer and H.H. Rugh: Transfer operators for coupled analytic maps, Ergodic Theory Dynam. Systems 20, (2000), pp 109 –143CrossRefGoogle Scholar
  23. 23.
    L. Gross: Decay of correlations in classical lattice models at high temperature, Comm. Math. Phys. 68, (1979), pp 9 –27CrossRefGoogle Scholar
  24. 24.
    X. Hu and S.J. Taylor: Fractal properties of products and projections of measures in math Rd, Math. Proc. Cambridge Philos. Soc. 115, (1994), pp 527 –544Google Scholar
  25. 25.
    B.R. Hunt and V.Yu. Kaloshin: How projections affect the dimension spectrum of fractal measures, Nonlinearity 10, (1997), pp 1031 –1046CrossRefGoogle Scholar
  26. 26.
    B.R. Hunt and V.Yu. Kaloshin: Regularity of embeddings of infinite-dimensonal fractal sets into finite-dimensional spaces, Nonlinearity 12, (1999), pp 1263 –1275CrossRefGoogle Scholar
  27. 27.
    B.R. Hunt, T. Sauer, and J.A. Yorke: Prevalence: a translation-invariant “almost every'' for infinite dimensional spaces, Bull. Am. Math. Soc. 27, (1992), pp 217 –238Google Scholar
  28. 28.
    T. Inoue: Sojourn times in small neighborhoods of indifferent fixed points of one-dimensional dynamical systems, Ergodic Theory Dynam. Systems 20, (2000), pp 241 –257CrossRefGoogle Scholar
  29. 29.
    R.B. Israel: High-temperature analyticity in classical lattice systems, Comm. Math. Phys. 50, (1976), pp 245 –257CrossRefGoogle Scholar
  30. 30.
    E. Järvenpää: A SRB-measure for globally coupled circle maps, Nonlinearity 10, (1997), pp 1435 –1469CrossRefGoogle Scholar
  31. 31.
    E. Järvenpää: A note on weakly coupled expanding maps on compact manifolds, Ann. Acad. Sci. Fenn. Math. 24, (1999), pp 511 –517Google Scholar
  32. 32.
    E. Järvenpää and M. Järvenpää: Linear mappings and generalized upper spectrum for dimensions, Nonlinearity 12, (1999), pp 475 –493CrossRefGoogle Scholar
  33. 33.
    E. Järvenpää and M. Järvenpää: On the definition of SRB-measures for coupled map lattices, Comm. Math. Phys. 220, (2001), pp 1 –12CrossRefGoogle Scholar
  34. 34.
    E. Järvenpää and T. Tolonen: Natural ergodic measures are not always observable, preprint 297, Scholar
  35. 35.
    M. Järvenpää: On the upper Minkowski dimension, the packing dimension, and orthogonal projections, Ann. Acad. Sci. Fenn. Math. Diss. 99, (1994), pp 1 –34Google Scholar
  36. 36.
    M. Jiang: Equilibrium states for lattice models of hyperbolic type, Nonlinearity 8, (1995), pp 631 –659CrossRefGoogle Scholar
  37. 37.
    M. Jiang and Ya.B. Pesin: Equilibrium measures for coupled map lattices: existence, uniqueness and finite-dimensional approximations, Comm. Math. Phys. 193, (1998), pp 675 –711CrossRefGoogle Scholar
  38. 38.
    W. Just: Globally coupled maps: phase transitions and synchronization, Phys. D 81, (1995), pp 317 –340CrossRefGoogle Scholar
  39. 39.
    W. Just: Bifurcations in globally coupled map lattices, J. Statist. Phys. 79, (1995), pp 429 –449CrossRefGoogle Scholar
  40. 40.
    K. Kaneko (ed): Theory and applications of coupled map lattices, Nonlinear Sci. Theory Appl. (John Wiley & Sons Ltd., Chichester 1993)Google Scholar
  41. 41.
    R. Kaufman: On Hausdorff dimension of projections, Mathematika 15, (1968), pp 153 –155.Google Scholar
  42. 42.
    R. Kaufman and P. Mattila: Hausdorff dimension and exceptional sets of linear transformations, Ann. Acad. Sci. Fenn. Ser. A I Math. 1, (1975), pp 387 –392Google Scholar
  43. 43.
    G. Keller:Completely mixing maps without limit measure, Colloq. Math. 100 (2004), no. 1, 73 –76.Google Scholar
  44. 44.
    G. Keller and M. Künzle: Transfer operators for coupled map lattices, Ergodic Theory Dynam. Systems 12, (1992), pp 297 –318Google Scholar
  45. 45.
    G. Keller and R. Zweimüller: Unidirectionally coupled interval maps: between dynamics and statistical mechanics, Nonlinearity 15, (2002), pp 1 –24CrossRefGoogle Scholar
  46. 46.
    M. Marstrand: Some fundamental geometrical properties of plane sets of fractional dimension, Proc. London Math. Soc. (3) 4, (1954), pp 257 –302Google Scholar
  47. 47.
    P. Mattila: Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Math. 1, (1975), pp 227 –244Google Scholar
  48. 48.
    P. Mattila: Hausdorff dimension and capacities of intersection of sets in $n$-space, Acta Math. 152, (1984), pp 77 –105Google Scholar
  49. 49.
    P. Mattila: Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability, (Cambridge University Press, Cambridge 1995)Google Scholar
  50. 50.
    P. Mattila: Hausdorff dimension, projections, and the Fourier transform, Publ. Math. 48, (2004), pp 3 –48Google Scholar
  51. 51.
    M. Misiurewicz and A. Zdunik: Convergence of images of certain measures. In: Statistical physics and dynamical systems (Köszeg, 1984), pp 203 –219, Progr. Phys. 10, (Birkhöuser Boston, Boston MA 1985)Google Scholar
  52. 52.
    Y. Peres and W. Schlag: Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102, (2000), pp 193 –251CrossRefGoogle Scholar
  53. 53.
    Ya.B. Pesin and Ya.G. Sinai: Space-time chaos in chains of weakly interacting hyperbolic mappings. Translated from the Russian by V.E. Nazaikinskii. Adv. Soviet Math. 3, In: Dynamical systems and statistical mechanics (Moscow, 1991), pp 165 –198, (Amer. Math. Soc., Providence RI 1991)Google Scholar
  54. 54.
    D. Ruelle: A measure associated with Axiom A attractors, Amer. J. Math. 98, (1976), pp 619 –654Google Scholar
  55. 55.
    D. Ruelle: Thermodynamic formalism. The mathematical structures of classical equilibrium statistical mechanics, Encyclopedia of Mathematics and its Applications 5 (Addison-Wesley Publishing Co., Reading Mass. 1978)Google Scholar
  56. 56.
    T.D. Sauer and J.A. Yorke: Are the dimensions of a set and its image equal under typical smooth functions?, Ergodic Theory Dynam. Systems 17, (1997), pp 941 –956CrossRefGoogle Scholar
  57. 57.
    M. Schmitt: Spectral theory for nonanalytic coupled map lattices, Nonlinearity 17, (2004), pp 671 –689CrossRefGoogle Scholar
  58. 58.
    B. Simon: The Statistical Mechanics of Lattice Gases, Vol 1, (Princeton University Press, Princeton New Jersey 1993)Google Scholar
  59. 59.
    Ya. Sinai: Gibbs measures in ergodic theory, Russian Math. Surveys 27, (1972), pp 21 –69Google Scholar
  60. 60.
    V. Volevich: Construction of an analogue of the Bowen-Ruelle-Sinai measure for a multidimensional lattice of interacting hyperbolic mappings (Russian), Mat. Sb. 184, (1993), pp 17 –36; translation in Russian Acad. Sci. Sb. Math. 79, (1994), pp 347 –363Google Scholar
  61. 61.
    L.-S. Young: What are SRB measures, and which dynamical systems have them?, J. Statist. Phys. 108, (2002), pp 733 –754.CrossRefGoogle Scholar

Authors and Affiliations

  • E Järvenpää
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläFinland

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