Multifractal analysis of nearly circular Julia set and thermodynamical formalism

P. Collet; R. Dobbertin; P. Moussa

Annales de l'I.H.P. Physique théorique (1992)

  • Volume: 56, Issue: 1, page 91-122
  • ISSN: 0246-0211

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Collet, P., Dobbertin, R., and Moussa, P.. "Multifractal analysis of nearly circular Julia set and thermodynamical formalism." Annales de l'I.H.P. Physique théorique 56.1 (1992): 91-122. <http://eudml.org/doc/76562>.

@article{Collet1992,
author = {Collet, P., Dobbertin, R., Moussa, P.},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {complex polynomial; Julia set; partition function; thermodynamic limit; entropy function; Hausdorff dimension; dimension spectrum; correlation dimension; Legendre transform; perturbation},
language = {eng},
number = {1},
pages = {91-122},
publisher = {Gauthier-Villars},
title = {Multifractal analysis of nearly circular Julia set and thermodynamical formalism},
url = {http://eudml.org/doc/76562},
volume = {56},
year = {1992},
}

TY - JOUR
AU - Collet, P.
AU - Dobbertin, R.
AU - Moussa, P.
TI - Multifractal analysis of nearly circular Julia set and thermodynamical formalism
JO - Annales de l'I.H.P. Physique théorique
PY - 1992
PB - Gauthier-Villars
VL - 56
IS - 1
SP - 91
EP - 122
LA - eng
KW - complex polynomial; Julia set; partition function; thermodynamic limit; entropy function; Hausdorff dimension; dimension spectrum; correlation dimension; Legendre transform; perturbation
UR - http://eudml.org/doc/76562
ER -

References

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  1. [1] M.H. Jensen, L.P. Kadanoff, A. Libchaber, I. Procaccia and J. Stavans, Phys. Rev. Lett., Vol. 55, 1985, pp. 2798-2801. 
  2. [2] T.C. Halsey, M.H. Jensen, L.P. Kadanoff, I. Procaccia and B.I. Shraiman, Phys. Rev., Vol. A 33, 1986, pp. 1141-1151. 
  3. [3] M.J. Feigenbaum, J. Stat. Phys., Vol. 46, 1987, pp. 919-924. Zbl0695.58021
  4. [4] M.J. Feigenbaum, M.H. Jensen and I. Procaccia, Phys. Rev. Lett., Vol. 57, 1986, pp. 1503-1506. MR858482
  5. [5] H.G.E. Hentschel and I. Procaccia, Physica, Vol. 8D, 1983, pp. 435-444. Zbl0538.58026MR719636
  6. [6] P. Grassberger and I. Procaccia, Physica, Vol. 13 D, 1984, pp. 34-54. Zbl0587.58031MR775277
  7. [7] R. Badii and A. Politi, J. Stat. Phys., Vol. 40, 1985, pp. 725-750. Zbl0627.58028MR806722
  8. [8] G. Paladin and A. Vulpiani, Lett. Nuovo Cim., Vol. 41, 1984, pp. 82-86; R. Benzi, G. Paladin, G. Parisi and A. Vulpiani, J. Phys. A: Math. Gen., Vol. 17, 1984, p. 3521-3531. MR770897
  9. [9] B.B. Mandelbrot, J. Fluid Mech., Vol. 62, 1974, pp. 331-358. Zbl0289.76031
  10. [10] T. Bohr and D. Rand, Physica, Vol. 25 D, 1987, pp. 387-398, D. Rand, Ergod. Theor. Dynam. Sys., Vol. 9, 1989, pp. 527-541. Zbl0643.58006MR887471
  11. [11] T. Bohr and T. Tel, The Thermodynamic of Fractals, in: Directions in Chaos, HAO BAI LIN Ed., World Scientific, Singapore, 1988, Vol. 2, pp. 194-237; T. Tel, Zeitschrift für Naturforschung, Vol. 43 a, 1988, pp. 1-72. 
  12. [12] G. Servizi, G. Turchetti and S. Vaienti, J. Phys. A: Math. Gen., Vol. 21, 1988, L 639-L 643. MR953440
  13. [13] S. Vaienti, J. Phys. A: Math. Gen., Vol. 21, 1988, pp. 2023-2043. Zbl0681.58026
  14. [14] J.P. Eckmann and I. Procaccia, Phys. Rev., Vol. A 34, 1986, pp. 659-661. 
  15. [15] O.E. Lanford, Entropy and equilibrium states in classical statistical mechanics, in: Statistical Mechanics and Mathematical Problems, A. LENARD Ed., Lecture Note in Physics, Vol. 20, pp. 1-113, Springer1973. 
  16. [16] P. Collet, J.L. Lebowitz and A. Porzio, J. Stat. Phys., Vol. 47, 1987, pp. 609-644. Zbl0683.58023MR912493
  17. [17] P. Blanchard, Bull. Am. Math. Soc., Vol. 11, 1984, pp. 85-141. Zbl0558.58017MR741725
  18. [18] D. RuelleErgod. Theor. Dynam. Sys., Vol. 2, 1982, pp. 99-107. Zbl0506.58024
  19. [19] M. Widom, D. Bensimon, L.P. Kadanoff and S.J. Shenker, J. Stat. Phys., Vol. 32, 1983, pp. 443-454. Zbl0594.58031MR725105
  20. [20] P. Collet, Hausdorff dimension of the singularities for invariant measures of expanding dynamical system in Dynamical Systems, Valparaiso1986, R. BAMÒN, R. LABARCA and J. PALIS Eds., Lecture Notes in Mathematics, Vol. 1331, pp. 47-58, Springer1988. Zbl0666.58026MR961092
  21. [21] P. MoussaHausdorff Dimension and Dimension Spectrum for Julia Sets Close to Unit Circle, in: Non Linear Dynamics, G. TURCHETTI Ed., World Scientific, Singapore, pp. 88-108, 1989; P. Moussa, Grandes déviations et analyse des objets fractals, in: Comptes Rendus des Rencontres entre Physiciens Théoriciens et Mathématiciens, R.C.P. 25, Vol. 41, pp. 131-142, Publ. de l'I.R.M.A., Université de Strasbourg, 1990. Zbl0741.30020
  22. [22] H. Brolin, Ark. Mat., Vol. 6, 1965, pp. 103-144. Zbl0127.03401MR194595
  23. [23] M.F. Barnsley, J.S. Geronimo and A.N. Harrington, Bull. Am. Math. Soc., Vol. 7, 1982, pp. 381-384. Zbl0509.30023MR663789
  24. [24] D. Sullivan, Seminar on Conformal and Hyperbolic Geometry, I.H.E.S. Preprint, Bures-sur-Yvette, France, 1982. 
  25. [25] R.S. Ellis, Entropy, Large Deviations and Statistical Mechanics, Springer, New York, 1985. Zbl0566.60097MR793553
  26. [26] S.R.S. Varadhan, Large Deviations and Applications, S.I.A.M. Regional Conference Series in Applied Mathematics, Vol. 46, 1984, see also: Grandes Déviations et Applications Statistiques, Séminaire, Faculté des Sciencesd'Orsay, 1977-1978, Astérisque, Vol. 68, 1978. MR758258
  27. [27] D. Plachky, Ann. Math. Stat., Vol. 42, 1971, pp. 1442-1443. Zbl0222.60016MR301844
  28. [28] D. Plachky and J. Steinebach, Period. Math., Vol. 6, 1974, pp. 338-340. 
  29. [29] D. Katzen and I. Procaccia, Phys. Rev. Lett., Vol. 58, 1987, pp. 1169-1172. MR879993
  30. [30] R. Badii and A. Politi, Phase Transitions in Hyperbolic Dynamical Systems, in: Chaos and Complexity, R. LIVI, S. RUFFO, S. CILIBERTO and M. BUIATTI Eds., World Scientific, Singapore, 1988, pp. 42-48; R. Badii, Rivista del Nuov. Cim., Vol. 12, No. 3, 1989, pp. 1-72. Zbl0794.58029MR970762
  31. [31] K.J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985. Zbl0587.28004MR867284
  32. [32] O. Frostman, Ann. Univ. Lund, 1935. 
  33. [33] J.P. Kahane, Ensembles Aléatoires et Dimensions, in: Recent Progress in Fourier Analysis, I. PERAL and J. L. RUBIO DE FRANCIA Ed., ElsevierNorth Holland, 1985. Zbl0596.60075MR848143
  34. [34] R.L. Dobrushin, Induction on Volume and No Cluster Expansion, in VIIIth International Congress On Mathematical Physics, M. MEBKHOUT and R. SENEOR Eds., World Scientific, Singapore, 1987, pp. 73-91. MR915563
  35. [35] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Springer Verlag, Berlin, Vol. 470, 1975. Zbl0308.28010MR442989
  36. [36] D. Ruelle, Thermodynamic formalism, Addison Wesley, Reading, 1978. MR511655
  37. [37] E. Vul, K. Khanin and Y. Sinai, Russ. Math. Surv., Vol. 39, 1984, pp. 1-40. Zbl0561.58033MR747790
  38. [38] G. Paladin and A. Vulpiani, Physics Reports, Vol. 156, 1987, pp. 147-225. MR919714
  39. [39] P. Billingsley, Ergodic Theory and Information, John Wiley & Sons, New York1965, pp. 136-145. Zbl0141.16702
  40. [40] A. Porzio, J. Stat. Phys., Vol. 58, 1990, pp. 923-937. Zbl0713.58031
  41. [41] G. Brown, G. Michon and J. Peyrière, On the Multifractal Analysis of Measures preprint, 1990. Zbl0892.28006

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