Acta Mathematica

Rigidity of escaping dynamics for transcendental entire functions

Lasse Rempe

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Abstract

We prove an analog of Böttcher’s theorem for transcendental entire functions in the Eremenko–Lyubich class $ \mathcal{B} $. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are quasiconformally equivalent in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points that remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal self-map of the plane.

We also prove that the conjugacy is essentially unique. In particular, we show that a function $ f \in \mathcal{B} $ has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic functions $ f,g \in \mathcal{B} $ that belong to the same parameter space are conjugate on their sets of escaping points.

Note

Supported in part by a postdoctoral fellowship of the German Academic Exchange Service (DAAD) and by EPSRC Advanced Research Fellowship EP/E052851/1.

Article information

Source
Acta Math., Volume 203, Number 2 (2009), 235-267.

Dates
Received: 11 March 2008
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892426

Digital Object Identifier
doi:10.1007/s11511-009-0042-y

Mathematical Reviews number (MathSciNet)
MR2570071

Zentralblatt MATH identifier
1226.37027

Subjects
Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]

Rights
2009 © Institut Mittag-Leffler

Citation

Rempe, Lasse. Rigidity of escaping dynamics for transcendental entire functions. Acta Math. 203 (2009), no. 2, 235--267. doi:10.1007/s11511-009-0042-y. https://projecteuclid.org/euclid.acta/1485892426


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References

  • A arts, J. M. & O versteegen, L. G., The geometry of Julia sets. Trans. Amer. Math. Soc., 338 (1993), 897–918.
  • A vila, A. & L yubich, M., Hausdorff dimension and conformal measures of Feigenbaum Julia sets. J. Amer. Math. Soc., 21 (2008), 305–363.
  • B arański, K., Trees and hairs for some hyperbolic entire maps of finite order. Math. Z., 257 (2007), 33–59.
  • B ergweiler, W., Iteration of meromorphic functions. Bull. Amer. Math. Soc., 29 (1993), 151–188.
  • B ergweiler, W., R ippon, P. J. & S tallard, G. M., Dynamics of meromorphic functions with direct or logarithmic singularities. Proc. Lond. Math. Soc., 97 (2008), 368–400.
  • B ers, L., The moduli of Kleinian groups. Uspeki Mat. Nauk, 29 (1974), 86–102 (Russian); English translation in Russian Math. Surveys, 29 (1974), 88–102.
  • B ers, L. & R oyden, H. L., Holomorphic families of injections. Acta Math., 157 (1986), 259–286.
  • B hattacharjee, R., D evaney, R. L., D eville, R. E. L., J osić, K. & M oreno-R ocha, M., Accessible points in the Julia sets of stable exponentials. Discrete Contin. Dyn. Syst. Ser. B, 1 (2001), 299–318.
  • D ouady, A. & G oldberg, L. R., The nonconjugacy of certain exponential functions, in Holomorphic Functions and Moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., 10, pp. 1–7. Springer, New York, 1988.
  • D ouady, A. & H ubbard, J. H., Étude dynamique des polynômes complexes. I, II. Publications Mathématiques d’Orsay, 84, 85. Université de Paris-Sud, Département de Mathématiques, Orsay, 1984, 1985.
  • — On the dynamics of polynomial-like mappings. Ann. Sci. École Norm. Sup., 18 (1985), 287–343.
  • E pstein, A. L., Towers of Finite Type Complex Analytic Maps. Ph.D. Thesis, City University of New York, 1995.
  • E remenko, A. È., On the iteration of entire functions, in Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publ., 23, pp. 339–345. PWN, Warsaw, 1989.
  • E remenko, A. È. & L yubich, M. Y. Structural stability in some families of entire functions. Preprint, Ukrainian Physico-Technical Institute, Kharkov, 1984 (Russian).
  • — Examples of entire functions with pathological dynamics. J. London Math. Soc., 36 (1987), 458–468.
  • — Dynamical properties of some classes of entire functions. Ann. Inst. Fourier (Grenoble), 42 (1992), 989–1020.
  • F atou, P., Sur l’itération des fonctions transcendantes entières. Acta Math., 47 (1926), 337–370.
  • G raczyk, J., Kotus, J. & Ś wiątek, G., Non-recurrent meromorphic functions. Fund. Math., 182 (2004), 269–281.
  • H ubbard, J. H., Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Vol. 1. Matrix Editions, Ithaca, NY, 2006.
  • J arque, X., On the connectivity of the escaping set for complex exponential Misiurewicz parameters. Preprint, 2009.
  • K ozlovski, O., S hen, W. & van S trien, S., Rigidity for real polynomials. Ann. of Math., 165 (2007), 749–841.
  • — Density of hyperbolicity in dimension one. Ann. of Math., 166 (2007), 145–182.
  • L angley, J. K. & Z heng, J. H., On the fixpoints, multipliers and value distribution of certain classes of meromorphic functions. Ann. Acad. Sci. Fenn. Math., 23 (1998), 133–150.
  • L ehto, O., An extension theorem for quasiconformal mappings. Proc. London Math. Soc., 14a (1965), 187–190.
  • L ehto, O. & V irtanen, K. I., Quasikonforme Abbildungen. Die Grundlehren der mathematischen Wissenschaften, 126. Springer, Berlin, 1965 (German); English translation: Quasiconformal Mappings in the Plane, Springer, New York, 1973.
  • M añé, R., S ad, P. & S ullivan, D., On the dynamics of rational maps. Ann. Sci. École Norm. Sup., 16 (1983), 193–217.
  • M cM ullen, C., Area and Hausdorff dimension of Julia sets of entire functions. Trans. Amer. Math. Soc., 300 (1987), 329–342.
  • Complex Dynamics and Renormalization. Annals of Mathematics Studies, 135. Princeton University Press, Princeton, NJ, 1994.
  • M ihaljević-B randt, H., Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds. Preprint, 2009. arXiv:0907.5398 [math.DS].
  • M ilnor, J., Dynamics in One Complex Variable. Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006.
  • R empe, L., Topological dynamics of exponential maps on their escaping sets. Ergodic Theory Dynam. Systems, 26:6 (2006), 1939–1975.
  • — On a question of Eremenko concerning escaping components of entire functions. Bull. Lond. Math. Soc., 39 (2007), 661–666.
  • — Siegel disks and periodic rays of entire functions. J. Reine Angew. Math., 624 (2008), 81–102.
  • — The escaping set of the exponential. To appear in Ergodic Theory Dynam. Systems.
  • — Connected escaping sets of exponential maps. Preprint, 2009. arXiv:0910.4680 [math.DS].
  • R empe, L. & van S trien, S., Absence of line fields and Mañé’s theorem for non-recurrent transcendental functions. To appear in Trans. Amer. Mat. Soc.
  • — Density of hyperbolicity for certain families of real transcendental entire functions. In preparation.
  • R ippon, P. J. & S tallard, G. M., Iteration of a class of hyperbolic meromorphic functions. Proc. Amer. Math. Soc., 127:11 (1999), 3251–3258.
  • R ottenfusser, G., R ückert, J., R empe, L. & S chleicher, D., Dynamic rays of bounded-type entire functions. To appear in Ann. of Math.
  • S hishikura, M., The boundary of the Mandelbrot set has Hausdorff dimension two, in Complex Analytic Methods in Dynamical Systems (Rio de Janeiro, 1992). Astérisque, 222 (1994), 7, 389–405.
  • van S trien, S., Misiurewicz maps unfold generically (even if they are critically non-finite). Fund. Math., 163 (2000), 39–54.
  • U rbański, M. & Z dunik, A., The finer geometry and dynamics of the hyperbolic exponential family. Michigan Math. J., 51 (2003), 227–250.
  • W einreich, J. M., Boundaries which Arise in the Iteration of Transcendental Entire Functions. Ph.D. Thesis, Imperial College, London, 1991.