[Hyperbolicité faible sur les orbites périodiques pour les polynômes]
On démontre que si les multiplicateurs des orbites périodiques répulsives d'un polynôme complexe croissent au moins comme n5+ε avec la période, où ε>0, alors l'ensemble de Julia du polynôme est localement connexe quand il est connexe. Comme conséquence on obtient que pour un polynôme complexe l'existence d'un cycle de Cremer implique l'existence d'une suite de cycles répulsifs ayant des multiplicateurs « petits ». D'une façon un peu surprenante la démonstration utilise des arguments de la théorie de la mesure.
We prove that if the multipliers of the repelling periodic orbits of a complex polynomial grow at least like n5+ε with the period, for some ε>0, then the Julia set of the polynomial is locally connected when it is connected. As a consequence for a polynomial the presence of a Cremer cycle implies the presence of a sequence of repelling periodic orbits with “small” multipliers. Somewhat surprisingly the proof is based on measure theorical considerations.
Publié le :
@article{CRMATH_2002__334_12_1113_0,
author = {Rivera-Letelier, J.},
title = {Weak hyperbolicity on periodic orbits for polynomials},
journal = {Comptes Rendus. Math\'ematique},
pages = {1113--1118},
publisher = {Elsevier},
volume = {334},
number = {12},
year = {2002},
doi = {10.1016/S1631-073X(02)02413-5},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02413-5/}
}
TY - JOUR AU - Rivera-Letelier, J. TI - Weak hyperbolicity on periodic orbits for polynomials JO - Comptes Rendus. Mathématique PY - 2002 SP - 1113 EP - 1118 VL - 334 IS - 12 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02413-5/ DO - 10.1016/S1631-073X(02)02413-5 LA - en ID - CRMATH_2002__334_12_1113_0 ER -
%0 Journal Article %A Rivera-Letelier, J. %T Weak hyperbolicity on periodic orbits for polynomials %J Comptes Rendus. Mathématique %D 2002 %P 1113-1118 %V 334 %N 12 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02413-5/ %R 10.1016/S1631-073X(02)02413-5 %G en %F CRMATH_2002__334_12_1113_0
Rivera-Letelier, J. Weak hyperbolicity on periodic orbits for polynomials. Comptes Rendus. Mathématique, Tome 334 (2002) no. 12, pp. 1113-1118. doi : 10.1016/S1631-073X(02)02413-5. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02413-5/
[1] Invariant sets under iteration of rational functions, Ark. Mat., Volume 6 (1965), pp. 103-144
[2] H. Bruin, S. van Strein, Expansion of derivatives in one dimensional dynamics, Preprint, September 2000
[3] On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup., Volume 18 (1985), pp. 287-344
[4] An invariant measure for rational maps, Bol. Soc. Brasil. Mat., Volume 14 (1983), pp. 45-62
[5] J. Graczyk, S. Smirnov, Weak expansion and geometry of Julia sets, March 1999 version
[6] M. Gromov, On the entropy of holomorphic maps, Preprint 1978
[7] Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynamical Systems, Volume 3 (1983), pp. 351-385
[8] Topological entropy and degree of smooth mappings, Bull. Acad. Polon. Sci., Volume 25 (1977), pp. 573-574
[9] The Hausdorff dimension of invariant probabilities of rational maps, Dynamical Systems, Valparaiso 1986, Lecture Notes in Math., 1331, Springer, 1988, pp. 86-117
[10] Absolutely continuous invariant measures under the summability condition, Invent. Math., Volume 105 (1991), pp. 123-136
[11] Sur les dynamiques holomorphes non linéarisables et une conjecture de V.I. Arnol'd, Ann. Sci. École Norm. Sup. (4), Volume 26 (1993), pp. 565-644
[12] Boundary Behavior of Conformal Maps, Springer-Verlag, Berlin, 1992
[13] Iterations of holomorphic Collet–Eckmann maps: Conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials, Trans. Amer. Math. Soc., Volume 350 (1998), pp. 717-742
[14] F. Przytycki, J. Rivera-Letelier, S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in iteration of rational maps, Preprint, 2000
[15] F. Przytycki, M. Urbański, Fractals in the Plane — the Ergodic Theory Methods, Cambridge University Press, to appear
[16] Porosity of Julia sets of non-recurrent and parabolic Collet Eckmann functions, Ann. Acad. Sci. Fenn. Math., Volume 26 (2001), pp. 125-154
[17] J. Rivera-Letelier, Rational maps with decay of geometry: Rigidity, Thurston's algorithm and local connectivity, Preprint IMS at Stony Brook #2000/9
[18] Conformal dynamical systems, Geometric Dynamics, Rio de Janeiro 1981, Lecture Notes in Math., 1007, Springer, 1983, pp. 727-752
[19] Petits diviseurs en dimension 1, Astérisque, Volume 231 (1995)
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