Multifractal analysis of nearly circular Julia set and thermodynamical formalism
Collet, P. ; Dobbertin, R. ; Moussa, P.
Annales de l'I.H.P. Physique théorique, Tome 56 (1992), p. 91-122 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{AIHPA_1992__56_1_91_0,
     author = {Collet, P. and Dobbertin, R. and Moussa, Pierre},
     title = {Multifractal analysis of nearly circular Julia set and thermodynamical formalism},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     volume = {56},
     year = {1992},
     pages = {91-122},
     mrnumber = {1149870},
     zbl = {0759.58033},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPA_1992__56_1_91_0}
}

Collet, P.; Dobbertin, R.; Moussa, P. Multifractal analysis of nearly circular Julia set and thermodynamical formalism. Annales de l'I.H.P. Physique théorique, Tome 56 (1992) pp. 91-122. http://gdmltest.u-ga.fr/item/AIHPA_1992__56_1_91_0/

[1] M.H. Jensen, L.P. Kadanoff, A. Libchaber, I. Procaccia and J. Stavans, Phys. Rev. Lett., Vol. 55, 1985, pp. 2798-2801.

[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] M.J. Feigenbaum, J. Stat. Phys., Vol. 46, 1987, pp. 919-924. | Zbl 0695.58021

[4] M.J. Feigenbaum, M.H. Jensen and I. Procaccia, Phys. Rev. Lett., Vol. 57, 1986, pp. 1503-1506. | MR 858482

[5] H.G.E. Hentschel and I. Procaccia, Physica, Vol. 8D, 1983, pp. 435-444. | MR 719636 | Zbl 0538.58026

[6] P. Grassberger and I. Procaccia, Physica, Vol. 13 D, 1984, pp. 34-54. | MR 775277 | Zbl 0587.58031

[7] R. Badii and A. Politi, J. Stat. Phys., Vol. 40, 1985, pp. 725-750. | MR 806722 | Zbl 0627.58028

[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. | MR 770897

[9] B.B. Mandelbrot, J. Fluid Mech., Vol. 62, 1974, pp. 331-358. | Zbl 0289.76031

[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. | MR 887471 | Zbl 0643.58006

[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] G. Servizi, G. Turchetti and S. Vaienti, J. Phys. A: Math. Gen., Vol. 21, 1988, L 639-L 643. | MR 953440

[13] S. Vaienti, J. Phys. A: Math. Gen., Vol. 21, 1988, pp. 2023-2043. | Zbl 0681.58026

[14] J.P. Eckmann and I. Procaccia, Phys. Rev., Vol. A 34, 1986, pp. 659-661.

[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, Springer 1973.

[16] P. Collet, J.L. Lebowitz and A. Porzio, J. Stat. Phys., Vol. 47, 1987, pp. 609-644. | MR 912493 | Zbl 0683.58023

[17] P. Blanchard, Bull. Am. Math. Soc., Vol. 11, 1984, pp. 85-141. | MR 741725 | Zbl 0558.58017

[18] D. Ruelle Ergod. Theor. Dynam. Sys., Vol. 2, 1982, pp. 99-107. | Zbl 0506.58024

[19] M. Widom, D. Bensimon, L.P. Kadanoff and S.J. Shenker, J. Stat. Phys., Vol. 32, 1983, pp. 443-454. | MR 725105 | Zbl 0594.58031

[20] P. Collet, Hausdorff dimension of the singularities for invariant measures of expanding dynamical system in Dynamical Systems, Valparaiso 1986, R. BAMÒN, R. LABARCA and J. PALIS Eds., Lecture Notes in Mathematics, Vol. 1331, pp. 47-58, Springer 1988. | MR 961092 | Zbl 0666.58026

[21] P. Moussa Hausdorff 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. | Zbl 0741.30020

[22] H. Brolin, Ark. Mat., Vol. 6, 1965, pp. 103-144. | MR 194595 | Zbl 0127.03401

[23] M.F. Barnsley, J.S. Geronimo and A.N. Harrington, Bull. Am. Math. Soc., Vol. 7, 1982, pp. 381-384. | MR 663789 | Zbl 0509.30023

[24] D. Sullivan, Seminar on Conformal and Hyperbolic Geometry, I.H.E.S. Preprint, Bures-sur-Yvette, France, 1982.

[25] R.S. Ellis, Entropy, Large Deviations and Statistical Mechanics, Springer, New York, 1985. | MR 793553 | Zbl 0566.60097

[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 Sciences d'Orsay, 1977-1978, Astérisque, Vol. 68, 1978. | MR 758258

[27] D. Plachky, Ann. Math. Stat., Vol. 42, 1971, pp. 1442-1443. | MR 301844 | Zbl 0222.60016

[28] D. Plachky and J. Steinebach, Period. Math., Vol. 6, 1974, pp. 338-340.

[29] D. Katzen and I. Procaccia, Phys. Rev. Lett., Vol. 58, 1987, pp. 1169-1172. | MR 879993

[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. | MR 970762 | Zbl 0794.58029

[31] K.J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985. | MR 867284 | Zbl 0587.28004

[32] O. Frostman, Ann. Univ. Lund, 1935.

[33] J.P. Kahane, Ensembles Aléatoires et Dimensions, in: Recent Progress in Fourier Analysis, I. PERAL and J. L. RUBIO DE FRANCIA Ed., Elsevier North Holland, 1985. | MR 848143 | Zbl 0596.60075

[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. | MR 915563

[35] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics, Springer Verlag, Berlin, Vol. 470, 1975. | MR 442989 | Zbl 0308.28010

[36] D. Ruelle, Thermodynamic formalism, Addison Wesley, Reading, 1978. | MR 511655

[37] E. Vul, K. Khanin and Y. Sinai, Russ. Math. Surv., Vol. 39, 1984, pp. 1-40. | MR 747790 | Zbl 0561.58033

[38] G. Paladin and A. Vulpiani, Physics Reports, Vol. 156, 1987, pp. 147-225. | MR 919714

[39] P. Billingsley, Ergodic Theory and Information, John Wiley & Sons, New York 1965, pp. 136-145. | Zbl 0141.16702

[40] A. Porzio, J. Stat. Phys., Vol. 58, 1990, pp. 923-937. | Zbl 0713.58031

[41] G. Brown, G. Michon and J. Peyrière, On the Multifractal Analysis of Measures preprint, 1990.