Skip to main content
SpringerLink
Log in

SRB-Measures for Coupled Map Lattices

  • Chapter
  • First Online:
Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 671))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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. V. Baladi: Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems 18, (1998), pp 255 –292

    Article  Google Scholar 

  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 –584

    Article  Google Scholar 

  4. V. Baladi and H.H. Rugh: Floquet spectrum of weakly coupled map lattices, Comm. Math. Phys. 220, (2001), pp 561 –582

    Google Scholar 

  5. J.-B. Bardet: Limit theorems for coupled analytic maps, Probab. Theory Related Fields 124, (2002), pp 151 –177

    Article  Google Scholar 

  6. M. Blank and L. Bunimovich: Multicomponent dynamical systems: SRB measures and phase transitions, Nonlinearity 16, (2003), pp 387 –401

    Article  Google Scholar 

  7. M. Blank, G. Keller, and C. Liverani: Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity 15, (2002), pp 1905 –1973

    Article  Google Scholar 

  8. R. Bowen: Equilibrium states and ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes in Math. 470, (1975).

    Google Scholar 

  9. J. Bricmont and A. Kupiainen: Coupled analytic maps, Nonlinearity 8, (1995), pp 379 –396

    Article  Google Scholar 

  10. J. Bricmont and A. Kupiainen: High temperature expansions and dynamical systems, Comm. Math. Phys. 178, (1996), pp 703 –732

    Google Scholar 

  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 –255

    Google Scholar 

  12. L.A. Bunimovich and Ya.G. Sinai: Spacetime chaos in coupled map lattices, Nonlinearity 1, (1988), pp 491 –516

    Article  Google Scholar 

  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 –91

    Google Scholar 

  14. R.L. Dobrushin: Gibbsian random fields for lattice systems with pairwise interactions, Funct. Anal. Appl. 2, (1968), pp 292 –301

    Article  Google Scholar 

  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 –224

    Article  Google Scholar 

  16. K.J. Falconer: Hausdorff dimension and the exceptional set of projections, Mathematika 29, (1982), pp 109 –115

    Google Scholar 

  17. K.J. Falconer: Geometry of Fractal Sets, (Cambridge University Press, Cambridge 1985)

    Google Scholar 

  18. K.J. Falconer and J.D. Howroyd: Projection theorems for box and packing dimensions, Math. Proc. Cambridge Philos. Soc. 119, (1996), pp 287 –295

    Google Scholar 

  19. K.J. Falconer and J.D. Howroyd: Packing dimensions of projections and dimension profiles, Math. Proc. Cambridge Philos. Soc. 121, (1997), pp 269 –286

    Article  Google Scholar 

  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 –713

    Google Scholar 

  21. K.J. Falconer and T.C. O'Neil: Convolutions and the geometry of multifractal measures, Math. Nachr. 204, (1999), pp 61 –82

    Google Scholar 

  22. T. Fischer and H.H. Rugh: Transfer operators for coupled analytic maps, Ergodic Theory Dynam. Systems 20, (2000), pp 109 –143

    Article  Google Scholar 

  23. L. Gross: Decay of correlations in classical lattice models at high temperature, Comm. Math. Phys. 68, (1979), pp 9 –27

    Article  Google Scholar 

  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 –544

    Google Scholar 

  25. B.R. Hunt and V.Yu. Kaloshin: How projections affect the dimension spectrum of fractal measures, Nonlinearity 10, (1997), pp 1031 –1046

    Article  Google Scholar 

  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 –1275

    Article  Google Scholar 

  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 –238

    Google Scholar 

  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 –257

    Article  Google Scholar 

  29. R.B. Israel: High-temperature analyticity in classical lattice systems, Comm. Math. Phys. 50, (1976), pp 245 –257

    Article  Google Scholar 

  30. E. Järvenpää: A SRB-measure for globally coupled circle maps, Nonlinearity 10, (1997), pp 1435 –1469

    Article  Google Scholar 

  31. E. Järvenpää: A note on weakly coupled expanding maps on compact manifolds, Ann. Acad. Sci. Fenn. Math. 24, (1999), pp 511 –517

    Google Scholar 

  32. E. Järvenpää and M. Järvenpää: Linear mappings and generalized upper spectrum for dimensions, Nonlinearity 12, (1999), pp 475 –493

    Article  Google Scholar 

  33. E. Järvenpää and M. Järvenpää: On the definition of SRB-measures for coupled map lattices, Comm. Math. Phys. 220, (2001), pp 1 –12

    Article  Google Scholar 

  34. E. Järvenpää and T. Tolonen: Natural ergodic measures are not always observable, preprint 297, http://www.math.jyu.fi/tutkimus/julkaisut.html

    Google Scholar 

  35. M. Järvenpää: On the upper Minkowski dimension, the packing dimension, and orthogonal projections, Ann. Acad. Sci. Fenn. Math. Diss. 99, (1994), pp 1 –34

    Google Scholar 

  36. M. Jiang: Equilibrium states for lattice models of hyperbolic type, Nonlinearity 8, (1995), pp 631 –659

    Article  Google Scholar 

  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 –711

    Article  Google Scholar 

  38. W. Just: Globally coupled maps: phase transitions and synchronization, Phys. D 81, (1995), pp 317 –340

    Article  Google Scholar 

  39. W. Just: Bifurcations in globally coupled map lattices, J. Statist. Phys. 79, (1995), pp 429 –449

    Article  Google Scholar 

  40. K. Kaneko (ed): Theory and applications of coupled map lattices, Nonlinear Sci. Theory Appl. (John Wiley & Sons Ltd., Chichester 1993)

    Google Scholar 

  41. R. Kaufman: On Hausdorff dimension of projections, Mathematika 15, (1968), pp 153 –155.

    Google Scholar 

  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 –392

    Google Scholar 

  43. G. Keller:Completely mixing maps without limit measure, Colloq. Math. 100 (2004), no. 1, 73 –76.

    Google Scholar 

  44. G. Keller and M. Künzle: Transfer operators for coupled map lattices, Ergodic Theory Dynam. Systems 12, (1992), pp 297 –318

    Google Scholar 

  45. G. Keller and R. Zweimüller: Unidirectionally coupled interval maps: between dynamics and statistical mechanics, Nonlinearity 15, (2002), pp 1 –24

    Article  Google Scholar 

  46. M. Marstrand: Some fundamental geometrical properties of plane sets of fractional dimension, Proc. London Math. Soc. (3) 4, (1954), pp 257 –302

    Google Scholar 

  47. P. Mattila: Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Math. 1, (1975), pp 227 –244

    Google Scholar 

  48. P. Mattila: Hausdorff dimension and capacities of intersection of sets in $n$-space, Acta Math. 152, (1984), pp 77 –105

    Google Scholar 

  49. P. Mattila: Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability, (Cambridge University Press, Cambridge 1995)

    Google Scholar 

  50. P. Mattila: Hausdorff dimension, projections, and the Fourier transform, Publ. Math. 48, (2004), pp 3 –48

    Google Scholar 

  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. Y. Peres and W. Schlag: Smoothness of projections, Bernoulli convolutions, and the dimension of exceptions, Duke Math. J. 102, (2000), pp 193 –251

    Article  Google Scholar 

  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. D. Ruelle: A measure associated with Axiom A attractors, Amer. J. Math. 98, (1976), pp 619 –654

    Google Scholar 

  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. 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 –956

    Article  Google Scholar 

  57. M. Schmitt: Spectral theory for nonanalytic coupled map lattices, Nonlinearity 17, (2004), pp 671 –689

    Article  Google Scholar 

  58. B. Simon: The Statistical Mechanics of Lattice Gases, Vol 1, (Princeton University Press, Princeton New Jersey 1993)

    Google Scholar 

  59. Ya. Sinai: Gibbs measures in ergodic theory, Russian Math. Surveys 27, (1972), pp 21 –69

    Google Scholar 

  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 –363

    Google Scholar 

  61. L.-S. Young: What are SRB measures, and which dynamical systems have them?, J. Statist. Phys. 108, (2002), pp 733 –754.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FIN-40014, Finland

    E Järvenpää

Authors
  1. E Järvenpää
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this chapter

Cite this chapter

Järvenpää, E. SRB-Measures for Coupled Map Lattices. In: Dynamics of Coupled Map Lattices and of Related Spatially Extended Systems. Lecture Notes in Physics, vol 671. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11360810_5

Download citation

Publish with us

Policies and ethics

Search

Navigation