Statistical properties of topological Collet-Eckmann maps

Przytycki, Feliks; Rivera-Letelier, Juan

doi:10.1016/j.ansens.2006.11.002

Przytycki, Feliks ; Rivera-Letelier, Juan

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 40 (2007) no. 1, pp. 135-178.

Zbl | 2 citations dans Numdam

DOI : 10.1016/j.ansens.2006.11.002

@article{ASENS_2007_4_40_1_135_0,
     author = {Przytycki, Feliks and Rivera-Letelier, Juan},
     title = {Statistical properties of topological {Collet-Eckmann} maps},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {135--178},
     publisher = {Elsevier},
     volume = {Ser. 4, 40},
     number = {1},
     year = {2007},
     doi = {10.1016/j.ansens.2006.11.002},
     zbl = {1115.37048},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.ansens.2006.11.002/}
}

TY  - JOUR
AU  - Przytycki, Feliks
AU  - Rivera-Letelier, Juan
TI  - Statistical properties of topological Collet-Eckmann maps
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2007
SP  - 135
EP  - 178
VL  - 40
IS  - 1
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.ansens.2006.11.002/
DO  - 10.1016/j.ansens.2006.11.002
LA  - en
ID  - ASENS_2007_4_40_1_135_0
ER  -

%0 Journal Article
%A Przytycki, Feliks
%A Rivera-Letelier, Juan
%T Statistical properties of topological Collet-Eckmann maps
%J Annales scientifiques de l'École Normale Supérieure
%D 2007
%P 135-178
%V 40
%N 1
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.ansens.2006.11.002/
%R 10.1016/j.ansens.2006.11.002
%G en
%F ASENS_2007_4_40_1_135_0

Przytycki, Feliks; Rivera-Letelier, Juan. Statistical properties of topological Collet-Eckmann maps. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 40 (2007) no. 1, pp. 135-178. doi : 10.1016/j.ansens.2006.11.002. http://archive.numdam.org/articles/10.1016/j.ansens.2006.11.002/

Bibliographie
Cité par

[1] Aspenberg M., The Collet-Eckmann condition for rational functions on the Riemann sphere, Ph.D. Thesis, KTH, Sweden, 2004.

[2] Beardon A.F., Iteration of Rational Functions. Complex Analytic Dynamical Systems, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. | MR | Zbl

[3] Bernard J., Dynamique des perturbations d'un exemple de Lattès, Ph.D. Thesis, Université de Paris-Sud, 1994.

[4] Bruin H., Keller G., Equilibrium states for S-unimodal maps, Ergodic Theory Dynam. Systems 18 (1998) 765-789. | MR | Zbl

[5] Bruin H., Luzzatto S., Van Strien S., Decay of correlations in one-dimensional dynamics, Ann. Sci. École Norm. Sup. 36 (2003) 621-646. | Numdam | MR | Zbl

[6] Bruin H., Todd M., Complex maps without invariant densities, Nonlinearity 19 (2006) 2929-2945. | MR | Zbl

[7] Carleson L., Gamelin T., Complex Dynamics, Springer-Verlag, Berlin/New York, 1993. | MR | Zbl

[8] Collet P., Eckmann J.P., Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems 3 (1983) 13-46. | MR | Zbl

[9] Denker M., Mauldin R.D., Nitecki Z., Urbański M., Conformal measures for rational functions revisited, Fund. Math. 157 (1998) 161-173. | MR | Zbl

[10] Denker M., Przytycki F., Urbanski M., On the transfer operator for rational functions on the Riemann sphere, Ergodic Theory Dynam. Systems 16 (1996) 255-266. | MR | Zbl

[11] Denker M., Urbański M., On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991) 365-384. | MR | Zbl

[12] Dinh T.C., Nguyen V.A., Sibony N., On thermodynamics of rational maps on the Riemann sphere, http://www.arxiv.org/math.DS/0603507. | MR

[13] Dobbs N., Critical points, cusps and induced expansion, Doctoral Thesis, Université Paris-Sud (Orsay), 2006.

[14] Dujardin R., Favre C., Distribution of rational maps with a preperiodic critical point, http://www.arxiv.org/math.DS/0601612.

[15] Gouëzel S., Vitesse de décorrélation et théorèmes limites pour les applications non uniformément dilatantes, Ph.D. Thesis, Université de Paris-Sud, 2004.

[16] Graczyk J., Smirnov S., Collet, Eckmann and Hölder, Invent. Math. 133 (1998) 69-96. | MR | Zbl

[17] Graczyk J., Smirnov S., Weak expansion and geometry of Julia sets, to appear in Invent. Math. A preprint version: March 1999.

[18] Graczyk J., Świa̧Tek G., Harmonic measure and expansion on the boundary of the connectedness locus, Invent. Math. 142 (2000) 605-629. | Zbl

[19] Haydn N., Convergence of the transfer operator for rational maps, Ergodic Theory Dynam. Systems 19 (1999) 657-669. | MR | Zbl

[20] Ledrappier F., Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynam. Systems 1 (1981) 77-93. | MR | Zbl

[21] Ledrappier F., Quelques propriétés ergodiques des applications rationnelles, C. R. Acad. Sci. Paris 299 (1984) 37-40. | MR | Zbl

[22] Keller G., Nowicki T., Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Comm. Math. Phys. 149 (1992) 31-69. | Zbl

[23] Mañé R., The Hausdorff dimension of invariant probabilities of rational maps, in: Dynamical Systems, Valparaiso, 1986, Lecture Notes in Math., vol. 1331, Springer-Verlag, Berlin, 1988, pp. 86-117. | MR | Zbl

[24] Martens M., Distortion results and invariant Cantor sets of unimodal mappings, Ergodic Theory Dynam. Systems 14 (1994) 331-349. | MR | Zbl

[25] Mcmullen C.T., Hausdorff dimension and conformal dynamics. II. Geometrically finite rational maps, Comment. Math. Helv. 75 (2000) 535-593. | MR | Zbl

[26] Mauldin R.D., Urbański M., Graph Directed Markov Systems. Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, vol. 148, Cambridge Univ. Press, Cambridge, 2003. | MR | Zbl

[27] Milnor J., Dynamics in One Complex Variable. Introductory Lectures, Friedr. Vieweg & Sohn, Braunschweig, 1999. | MR | Zbl

[28] Nowicki T., Sands D., Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math. 132 (1998) 633-680. | MR | Zbl

[29] Pesin Y., Senti S., Equilibrium measures for some one dimensional maps, Mosc. Math. J. 5 (2005) 669-678. | MR | Zbl

[30] Przytycki F., Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993) 309-317. | MR | Zbl

[31] Przytycki F., On measure and Hausdorff dimension of Julia sets for holomorphic Collet-Eckmann maps, in: Ledrappier F., Lewowicz J., Newhouse S. (Eds.), Int. Conf. on Dynamical Systems-a tribute to R. Mañé, Montevideo, 1995, Pitman Res. Notes in Math. Series, vol. 362, Longman, Harlow, 1996, pp. 167-181. | Zbl

[32] Przytycki F., Iterations of holomorphic Collet-Eckmann maps: Conformal and invariant measures, Trans. Amer. Math. Soc. 350 (1998) 717-742. | Zbl

[33] Przytycki F., Hölder implies CE, Astérisque 261 (2000) 385-403. | Numdam | MR | Zbl

[34] Przytycki F., Conical limit set and Poincaré exponent for iterations of rational functions, Trans. Amer. Math. Soc. 351 (1999) 2081-2099. | MR | Zbl

[35] Przytycki F., Rivera-Letelier J., Smirnov S., Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math. 151 (2003) 29-63. | MR | Zbl

[36] Przytycki F., Rohde S., Rigidity of holomorphic Collet-Eckmann repellers, Ark. Mat. 37 (1999) 357-371. | Zbl

[37] Przytycki F., Urbański M., Fractals in the Plane, Ergodic Theory Methods, Cambridge Univ. Press, in press. Available on, http://www.math.unt.edu/~urbanski.

[38] Urbański M., Measures and dimensions in conformal dynamics, Bull. Amer. Math. Soc. 40 (2003) 281-321. | MR | Zbl

[39] Rees M., Positive measure sets of ergodic rational maps, Ann. Sci. École Norm. Sup. 19 (1986) 383-407. | Numdam | MR | Zbl

[40] Rivera-Letelier J., A connecting lemma for rational maps satisfying a no-growth condition, Ergodic Theory Dynam. Systems 27 (2007) 595-636. | MR | Zbl

[41] Senti S., Dimension of weakly expanding points for quadratic maps, Bull. Soc. Math. France 131 (2003) 399-420. | Numdam | MR | Zbl

[42] Smirnov S., Symbolic dynamics and Collet-Eckmann condition, Internat. Math. Res. Notices 7 (2000) 333-351. | Zbl

[43] Sullivan D., Conformal dynamical systems, in: Geometric Dynamics, Rio de Janeiro, 1981, Lecture Notes in Math., vol. 1007, Springer-Verlag, Berlin, 1983, pp. 725-752. | MR | Zbl

[44] Urbański M., Measures and dimensions in conformal dynamics, Bull. Amer. Math. Soc. 40 (2003) 281-321. | MR | Zbl

[45] Young L.-S., Decay of correlations for certain quadratic maps, Comm. Math. Phys. 146 (1992) 123-138. | MR | Zbl

[46] Young L.-S., Recurrence times and rates of mixing, Israel J. Math. 110 (1999) 153-188. | MR | Zbl

[47] Zdunik A., Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990) 627-649. | MR | Zbl

Cité par Sources :