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

Benini, Anna Miriam; Lyubich, Mikhail

Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1493-1520.

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.

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.

Classification : 37F10, 37F20
Keywords: rigidity, accessibility, exponential maps, combinatorics
Mot clés : rigidité, atterrissage de rayons, fonction exponentielle, combinatorique

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

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

[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., Volume 9 (1999) no. 8, pp. 1517-1534 | DOI | MR | Zbl

[4] Devaney, Robert L.; Krych, Michał Dynamics of $\exp \left(z\right)$, Ergodic Theory Dynam. Systems, Volume 4 (1984) no. 1, pp. 35-52 | DOI | MR | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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