Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T16:31:07.975Z Has data issue: false hasContentIssue false
  • English
  • Français

Parameter rays in the space of exponential maps

Published online by Cambridge University Press:  01 April 2009

MARKUS FÖRSTER  and
DIERK SCHLEICHER
MARKUS FÖRSTER
Affiliation:
School of Engineering and Science, Research I, Jacobs University, Formerly International University Bremen, Postfach 750 561, D-28725 Bremen, Germany (email: [email protected], [email protected])
DIERK SCHLEICHER
Affiliation:
School of Engineering and Science, Research I, Jacobs University, Formerly International University Bremen, Postfach 750 561, D-28725 Bremen, Germany (email: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the set I of parameters κ∈ℂ for which the singular orbit (0,eκ,…) of Eκ(z):=exp (z+κ) converges to . These parameters are organized in curves in parameter space called parameter rays, together with endpoints of certain rays. Parameter rays are an important tool to understand the detailed structure of exponential parameter space. In this paper, we construct and investigate these parameter rays. Based on these results, a complete classification of the set I is given in the following paper [M. Förster, L. Rempe and D. Schleicher. Classification of escaping exponential maps. Proc. Amer. Math. Soc.136 (2008), 651–663].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 2 , April 2009 , pp. 515 - 544
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Avila, A., Kahn, J., Lyubich, M. and Shen, W.. Combinatorial rigidity for unicritical polynomials. Preprint 05, Stony Brook, 2005.Google Scholar
[2]Bailesteanu, M., Balan, V. and Schleicher, D.. Hausdorff dimension of exponential parameter rays and their endpoints. Nonlinearity 21 (2008), 113–130.CrossRefGoogle Scholar
[3]Baker, N. and Rippon, P.. Iteration of exponential functions. Ann. Acad. Sci. Fenn. Ser. A. I. Math. 9 (1984), 49–77.CrossRefGoogle Scholar
[4]Bodelón, C., Devaney, R., Goldberg, L., Hayes, M., Hubbard, J. and Roberts, G.. Hairs for the complex exponential family. Internat. J. Bifur. Chaos 9(8) (1999), 1517–1534.CrossRefGoogle Scholar
[5]Bodelón, C., Devaney, R., Goldberg, L., Hayes, M., Hubbard, J. and Roberts, G.. Dynamical convergence of polynomials to the exponential. J. Difference Equ. Appl. 6(3) (2000), 275–307.CrossRefGoogle Scholar
[6]Devaney, R. and Krych, M.. Dynamics of exp (z). Ergod. Th. & Dynam. Sys. 4 (1984), 35–52.CrossRefGoogle Scholar
[7]Devaney, R. and Tangerman, F.. Dynamics of entire functions near the essential singularity. Ergod. Th. & Dynam. Sys. 6 (1986), 489–503.CrossRefGoogle Scholar
[8]Douady, A. and Hubbard, J.. Etude dynamique des polynômes complexes. Prépubl. Math. d’Orsay 84-02, 1984 (part 1) and 85-04, 1985 (part 2).Google Scholar
[9]Eremenko, A. and Lyubich, M.. Dynamical properties of some classes of entire functions. Ann. Sci. Inst. Fourier, Grenoble 42(4) (1992), 989–1020.CrossRefGoogle Scholar
[10]Fagella, N.. Limiting dynamics for the complex standard family. Internat. J. Bifur. Chaos 5(3) (1995), 673–699.CrossRefGoogle Scholar
[11]Förster, M.. Parameter rays for the exponential family. Diploma Thesis, Technische Universität München, 2003.Google Scholar
[12]Förster, M.. Exponential maps with escaping singular orbits. PhD Thesis, International University, Bremen, 2006.Google Scholar
[13]Förster, M., Rempe, L. and Schleicher, D.. Classification of escaping exponential maps. Proc. Amer. Math. Soc. 136 (2008), 651–663.CrossRefGoogle Scholar
[14]Hubbard, J.. Local connectivity of bifurcation loci: three theorems of Jean-Christophe Yoccoz. Topological Methods in Modern Mathematics. Publish or Perish, Houston, TX, 1993, pp. 375–378 and 467–511.Google Scholar
[15]Karpińska, B.. Hausdorff dimension of the hairs without endpoints for λexp (z). C. R. Acad. Sci. Paris, Sér. I 328 (1999), 1039–1044.CrossRefGoogle Scholar
[16]Keen, L. and Kotus, J.. Dynamics of the family λtan z. J. Conf. Geometry Dynamics 1(4) (1997), 28–57.CrossRefGoogle Scholar
[17]Laubner, B., Schleicher, D. and Vicol, V.. A dynamical classification of postsingularly finite exponential maps. Discrete Contin. Dyn. Syst. 22(3) (2008), 663–682.CrossRefGoogle Scholar
[18]Qiu, W.. Hausdorff dimension of the M-set of λexp (z). Acta Math. Sinica (N.S.) 10(4) (1994), 362–368.Google Scholar
[19]Rempe, L.. Dynamics of exponential maps. Dissertation Thesis, Christian-Albrechts-Universität, Kiel, 2003.Google Scholar
[20]Rempe, L.. Topological dynamics of exponential maps on their escaping sets. Ergod. Th. & Dynam. Sys. 26 (2006), 1939–1975.CrossRefGoogle Scholar
[21]Rempe, L.. A landing theorem for periodic rays of exponential maps. Proc. Amer. Math. Soc. 134(9) (2006), 2639–2648.CrossRefGoogle Scholar
[22]Rempe, L. and Schleicher, D.. Bifurcations in the space of exponential maps. Invent. Math. to appear, DOI: 10.1007/s00222-008-0147-5.Google Scholar
[23]Rempe, L. and Schleicher, D.. Combinatorics of bifurcations in exponential parameter space. Transcendental Dynamics and Complex Analysis (London Mathematical Society Lecture Note Series). Eds. P. Rippon and G. Stallard. Cambridge University Press, Cambridge, 2007, pp. 317–370.Google Scholar
[24]Rempe, L. and Schleicher, D.. Bifurcation loci of exponential maps and quadratic polynomials: local connectivity, triviality of fibers, and density of hyperbolicity. Holomorphic Dynamics and Renormalization Theory, Volume in Honor of John Milnor’s 75th Birthday (Fields Institute Communications, 53). Eds. M. Lyubich and M. Yampolski. Fields Institute, Toronto, 2008, to appear.CrossRefGoogle Scholar
[25]Rottenfußer, G.. On the dynamical fine structure of entire transcendental functions. PhD Thesis, International University, Bremen, 2005.Google Scholar
[26]Rottenfußer, G., Rückert, J., Rempe, L. and Schleicher, D.. Dynamic rays of bounded-type entire functions. Preprint, 2007, ArXiv 0704.3213.Google Scholar
[27]Rottenfußer, G. and Schleicher, D.. Escaping points of the cosine family. Transcendental Dynamics and Complex Analysis (London Mathematical Society Lecture Note Series). Eds. P. Rippon and G. Stallard. Cambridge University Press, Cambridge, 2008, ArXiv math.DS/0403012.Google Scholar
[28]Schleicher, D.. On the dynamics of iterated exponential maps. Habilitation Thesis, TU München, 1999.Google Scholar
[29]Schleicher, D.. On fibers and local connectivity of Mandelbrot and multibrot sets. Fractal Geometry and Applications: a Jubilee of Benoǐt Mandelbrot (Proceedings of Symposia in Pure Mathematics, 72, Part I). Eds. M. L. Lapidus et al. American Mathematical Society, Providence, RI, 2004, pp. 477–517.CrossRefGoogle Scholar
[30]Schleicher, D.. Attracting dynamics of exponential maps. Ann. Acad. Sci. Fenn. Math. 28 (2003), 3–34.Google Scholar
[31]Schleicher, D.. Hyperbolic components in exponential parameter space. C. R. Acad. Sci. Paris, Sér. I 339 (2004), 223–228.CrossRefGoogle Scholar
[32]Schleicher, D.. Hausdorff dimension, its properties, and its surprises. Amer. Math. Monthly 114(6) (2007), 509–528.CrossRefGoogle Scholar
[33]Schleicher, D.. The dynamical fine structure of iterated cosine maps and a dimension paradox. Duke Math. J. 136(2) (2007), 343–356.CrossRefGoogle Scholar
[34]Schleicher, D. and Zimmer, J.. Escaping points of exponential maps. J. London Math. Soc. 67 (2003), 380–400.CrossRefGoogle Scholar
[35]Schleicher, D. and Zimmer, J.. Periodic points and dynamic rays of exponential maps. Ann. Acad. Sci. Fenn. 28(2) (2003), 327–354.Google Scholar
[36]Viana da Silva, M.. The differentiability of the hairs of exp (z). Proc. Amer. Math. Soc. 103(4) (1988), 1179–1184.Google Scholar
[37]Ye, Z.. Structural instability of exponential functions. Trans. Amer. Math. Soc. 344(1) (1994), 379–389.CrossRefGoogle Scholar