Repelling periodic points and landing of rays for post-singularly bounded exponential maps

[Points périodiques répulsifs et atterrissage de rayons pour les fonctions exponentielles avec orbite singulière limitée]

Benini, Anna Miriam ; Lyubich, Mikhail

Annales de l'Institut Fourier, Tome 64 (2014), p. 1493-1520 / Harvested from Numdam

Résumé

On montre que, pour toute fonction exponentielle avec orbite singulière bornée, chaque point périodique répulsif est le point d’atterrissage d’au moins un rayon externe avec adresse périodique. Cela généralise un théorème de Douady qui s’applique au polynômes complexes avec ensemble de Julia connexe, dont on donne une nouvelle démonstration.

Abstract

We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present a new proof. In both cases we also show that points in hyperbolic sets are accessible by at least one and at most finitely many rays. For exponentials this allows us to conclude that the singular value itself is accessible.

Détail

Publié le : 2014-01-01

MR 3329671 | Zbl 06387315 | Zbl 1323.37029

DOI : https://doi.org/10.5802/aif.2888
Classification:  37F10,  37F20
Mots clés: rigidité, atterrissage de rayons, fonction exponentielle, combinatorique

Bibliographie

[1] Baker, I. N.; Rippon, P. J. Iteration of exponential functions, Ann. Acad. Sci. Fenn. Ser. A I Math., Tome 9 (1984), pp. 49-77 | Article | MR 752391 | Zbl 0558.30029

[2] Benini, Anna Miriam Expansivity properties and rigidity for non-recurrent exponential maps (arXiv:1210.1353 [math.DS]) | MR 3366634 | Zbl 1359.37102 | Zbl 06476843

[3] Bodelón, Clara; Devaney, Robert L.; Hayes, Michael; Roberts, Gareth; Goldberg, Lisa R.; Hubbard, John H. Hairs for the complex exponential family, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Tome 9 (1999) no. 8, pp. 1517-1534 | Article | MR 1721835 | Zbl 1192.37061

[4] Devaney, Robert L.; Krych, Michał Dynamics of ${\rm exp}(z)$, Ergodic Theory Dynam. Systems, Tome 4 (1984) no. 1, pp. 35-52 | Article | MR 758892 | Zbl 0567.58025

[5] Erëmenko, A. È.; Levin, G. M. Periodic points of polynomials, Ukrain. Mat. Zh., Tome 41 (1989) no. 11, p. 1467-1471, 1581 | Article | MR 1036486 | Zbl 0686.30019

[6] Erëmenko, A. È.; Lyubich, M. Yu. Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble), Tome 42 (1992) no. 4, pp. 989-1020 | Article | Numdam | MR 1196102 | Zbl 0735.58031

[7] Förster, Markus; Schleicher, Dierk Parameter rays in the space of exponential maps, Ergodic Theory Dynam. Systems, Tome 29 (2009) no. 2, pp. 515-544 | Article | MR 2486782 | Zbl 1160.37364

[8] Hubbard, J. H. Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX (1993), pp. 467-511 | MR 1215974 | Zbl 0797.58049

[9] Ljubich, M. Ju. Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, Tome 3 (1983) no. 3, pp. 351-385 | Article | MR 741393 | Zbl 0537.58035

[10] Makienko, P.; Sienra, G. Poincaré series and instability of exponential maps, Bol. Soc. Mat. Mexicana (3), Tome 12 (2006) no. 2, pp. 213-228 | MR 2292985 | Zbl 1219.37035

[11] Milnor, John Dynamics in one complex variable, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 160 (2006), pp. viii+304 | MR 2193309 | Zbl 1085.30002

[12] Przytycki, F. Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps, Fund. Math., Tome 144 (1994) no. 3, pp. 259-278 | MR 1279481 | Zbl 0812.58058

[13] Rempe, Lasse A landing theorem for periodic rays of exponential maps, Proc. Amer. Math. Soc., Tome 134 (2006) no. 9, p. 2639-2648 (electronic) | Article | MR 2213743 | Zbl 1099.37040

[14] Rempe, Lasse Topological dynamics of exponential maps on their escaping sets, Ergodic Theory Dynam. Systems, Tome 26 (2006) no. 6, pp. 1939-1975 | Article | MR 2279273 | Zbl 1138.37025

[15] Rempe, Lasse On nonlanding dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., Tome 32 (2007) no. 2, pp. 353-369 | MR 2337482 | Zbl 1116.37033

[16] Rempe, Lasse; Schleicher, Dierk Combinatorics of bifurcations in exponential parameter space, Transcendental dynamics and complex analysis, Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 348 (2008), pp. 317-370 | Article | MR 2458808 | Zbl 1178.37043

[17] Rempe, Lasse; Van Strien, Sebastian Absence of line fields and Mañé’s theorem for nonrecurrent transcendental functions, Trans. Amer. Math. Soc., Tome 363 (2011) no. 1, pp. 203-228 | Article | MR 2719679 | Zbl 1230.37056

[18] Schleicher, Dierk On fibers and local connectivity of Mandelbrot and Multibrot sets, Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 72 (2004), pp. 477-517 | MR 2112117 | Zbl 1074.30025

[19] Schleicher, Dierk; Zimmer, Johannes Escaping points of exponential maps, J. London Math. Soc. (2), Tome 67 (2003) no. 2, pp. 380-400 | Article | MR 1956142 | Zbl 1042.30012

[20] Schleicher, Dierk; Zimmer, Johannes Periodic points and dynamic rays of exponential maps, Ann. Acad. Sci. Fenn. Math., Tome 28 (2003) no. 2, pp. 327-354 | MR 1996442 | Zbl 1088.30017