Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T13:18:03.215Z Has data issue: false hasContentIssue false
  • English
  • Français

Rigidity and absence of line fields for meromorphic and Ahlfors islands maps

Published online by Cambridge University Press:  23 November 2011

VOLKER MAYER  and
LASSE REMPE
VOLKER MAYER
Affiliation:
Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS, 59655 Villeneuve d’Ascq Cedex, France (email: volker.mayer@math.univ-lille1.fr)
LASSE REMPE
Affiliation:
Department of Mathematical Sciences, University of Liverpool, L69 7ZL, UK (email: l.rempe@liverpool.ac.uk)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we give an elementary proof of the absence of invariant line fields on the conical Julia set of an analytic function of one variable. This proof applies not only to rational and transcendental meromorphic functions (where it was previously known), but even to the extremely general setting of Ahlfors islands maps as defined by Epstein. In fact, we prove a more general result on the absence of invariant differentials, measurable with respect to a conformal measure that is supported on the (unbranched) conical Julia set. This includes the study of cohomological equations for log ∣f′∣, which are relevant to a number of well-known rigidity questions. In particular, we prove the absence of continuous line fields on the Julia set of any transcendental entire function.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 5 , October 2012 , pp. 1691 - 1710
Copyright
Copyright © Cambridge University Press 2011

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

[Ba]Baker, I. N.. Repulsive fixed points of entire functions. Math. Z. 104 (1968), 252–256.CrossRefGoogle Scholar
[BDH]Baker, I. N., Domínguez, P. and Herring, M. E.. Dynamics of functions meromorphic outside a small set. Ergod. Th. & Dynam. Sys. 21(3) (2001), 647–672.CrossRefGoogle Scholar
[BFU]Bedford, T., Fisher, A. and Urbański, M.. The scenery flow for hyperbolic Julia sets. Proc. Lond. Math. Soc. 3(85) (2002), 467–492.CrossRefGoogle Scholar
[Be]Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151–188.CrossRefGoogle Scholar
[BwEr]Bergweiler, W. and Eremenko, A.. Meromorphic functions with linearly distributed values and Julia sets of rational functions. Proc. Amer. Math. Soc. 137 (2009), 2329–2333.CrossRefGoogle Scholar
[Ep1]Epstein, A.. Towers of finite type complex analytic maps. PhD Thesis, City University of New York, 1995.Google Scholar
[Ep2]Epstein, A.. Dynamics of finite type complex analytic maps I: global structure theory, unpublished.Google Scholar
[EO]Epstein, A. and Oudkerk, R.. Iteration of Ahlfors and Picard functions which overflow their domains, unpublished.Google Scholar
[EvS]Eremenko, A. and van Strien, S.. Rational maps with real multipliers. Trans. Amer. Math. Soc. 363 (2011), 6453–6463.CrossRefGoogle Scholar
[FU]Fisher, A. and Urbański, M.. On invariant line fields. Bull. Lond. Math. Soc. 32 (2000), 555–570.CrossRefGoogle Scholar
[GKŚ]Graczyk, J., Kotus, J. and Świątek, G.. Non-recurrent meromorphic functions. Fund. Math. 182(3) (2004), 269–281.CrossRefGoogle Scholar
[H]Haïssinsky, P.. Rigidity and expansion for rational maps. J. Lond. Math. Soc. (2) 63 (2001), 128–140.CrossRefGoogle Scholar
[KU]Kotus, J. and Urbański, M.. The class of pseudo non-recurrent elliptic functions; geometry and dynamics. Preprint, 2007.Google Scholar
[L]Ledrappier, F.. Quelques propriétés ergodiques des applications rationnelles. C. R. Acad. Sci. Paris Sér. I Math. 299(1) (1984), 37–40.Google Scholar
[Mk]Makarov, N. G.. On the distortion of boundary sets under conformal mappings. Proc. Lond. Math. Soc. 51 (1985), 369–384.CrossRefGoogle Scholar
[MSS]Mané, R., Sad, P. and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4) 16 (1983), 193–217.CrossRefGoogle Scholar
[MM]Martin, G. J. and Mayer, V.. Rigidity in holomorphic and quasiregular dynamics. Trans. Amer. Math. Soc. 355(11) (2003), 4297–4347.CrossRefGoogle Scholar
[My]Mayer, V.. Comparing measures and invariant line fields. Ergod. Th. & Dynam. Sys. 22 (2002), 555–570.CrossRefGoogle Scholar
[MyUr]Mayer, V. and Urbański, M.. Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order. Mem. Amer. Math. Soc. 203 (2010), 954.Google Scholar
[McM1]McMullen, C. T.. Complex Dynamics and Renormalization (Annals of Mathematics Studies, 135). Princeton University Press, Princeton, NJ, 1994.Google Scholar
[McM2]McMullen, C. T.. The Mandelbrot set is universal. The Mandelbrot Set, Theme and Variations (London Mathematical Society Lecture Note Series, 274). Cambridge University Press, Cambridge, 2000, pp. 1–17.Google Scholar
[N]Nevanlinna, R.. Eindeutige analytische Funktionen. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd XLVI. Springer, Berlin, 1953, 2te Aufl.Google Scholar
[P]Przytycki, F.. Conical limit sets and Poincare exponent for iterations of rational functions. Trans. Amer. Math. Soc. 351.5 (1999), 2081–2099.CrossRefGoogle Scholar
[PR]Przytycki, F. and Rhode, S.. Rigidity of holomorphic Collet–Eckmann repellers. Ark. Mat. 37 (1999), 357–371.CrossRefGoogle Scholar
[PU]Przytycki, F. and Urbański, M.. Rigidity of tame rational functions. Bull. Pol. Acad. Sci. Math. 47.2 (1999), 163–182.Google Scholar
[Re]Rempe, L.. Hyperbolic Dimension and radial Julia sets of transcendental functions. Proc. Amer. Math. Soc. 137(4) (2009), 1411–1420.CrossRefGoogle Scholar
[RvS]Rempe, L. and van Strien, S.. Absence of line fields and Mañé’s theorem for non-recurrent transcendental functions. Trans. Amer. Math. Soc. 363 (2011), 203–228.CrossRefGoogle Scholar
[RR]Rempe, L. and Rippon, P.. Exotic Baker and wandering domains for Ahlfors islands functions. Preprint, 2010, arXiv:1008.1724.Google Scholar
[Su1]Sullivan, D.. Quasiconformal homeomorphisms in dynamics, topology and geometry. Proc. Internat. Congress of Math.. American Mathematical Society, Berkeley, 1986, pp. 1216–1228.Google Scholar
[Su2]Sullivan, D.. Conformal dynamical systems. Geometric Dynamics (Lecture Notes in Mathematics, 1007). Springer, New York, p. 752–52.Google Scholar
[Ur]Urbański, M.. Geometric rigidity for class S of transcendental meromorphic functions whose Julia sets are Jordan curves. Proc. Amer. Math. Soc. 137 (2009), 3733–3739.CrossRefGoogle Scholar
[Zd]Zdunik, A.. Parabolic orbifolds and the dimension of maximal measure for rational maps. Invent. Math. 99 (1990), 627–649.CrossRefGoogle Scholar