Comptes Rendus
Weak hyperbolicity on periodic orbits for polynomials
[Hyperbolicité faible sur les orbites périodiques pour les polynômes]
Comptes Rendus. Mathématique, Volume 334 (2002) no. 12, pp. 1113-1118.

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.

Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02413-5

J. Rivera-Letelier 1

1 Institute for Mathematical Sciences SUNY, Stony Brook, NY 11794-3651, USA
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J. Rivera-Letelier. Weak hyperbolicity on periodic orbits for polynomials. Comptes Rendus. Mathématique, Volume 334 (2002) no. 12, pp. 1113-1118. doi : 10.1016/S1631-073X(02)02413-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02413-5/

[1] H. Brolin 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] A. Douady; J. Hubbard On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup., Volume 18 (1985), pp. 287-344

[4] A. Freire; A. Lopes; R. Mañé 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] M. Ljubich Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynamical Systems, Volume 3 (1983), pp. 351-385

[8] M. Misiurewicz; F. Przytycki Topological entropy and degree of smooth mappings, Bull. Acad. Polon. Sci., Volume 25 (1977), pp. 573-574

[9] R. Mañé The Hausdorff dimension of invariant probabilities of rational maps, Dynamical Systems, Valparaiso 1986, Lecture Notes in Math., 1331, Springer, 1988, pp. 86-117

[10] T. Nowicki; S. van Strein Absolutely continuous invariant measures under the summability condition, Invent. Math., Volume 105 (1991), pp. 123-136

[11] R. Perez-Marco 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] Ch. Pommerenke Boundary Behavior of Conformal Maps, Springer-Verlag, Berlin, 1992

[13] F. Przytycki 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] F. Przytycki; M. Urbański 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] D. Sullivan Conformal dynamical systems, Geometric Dynamics, Rio de Janeiro 1981, Lecture Notes in Math., 1007, Springer, 1983, pp. 727-752

[19] J.C. Yoccoz Petits diviseurs en dimension 1, Astérisque, Volume 231 (1995)

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