Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-30T01:18:29.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant measures for meromorphic Misiurewicz maps

Published online by Cambridge University Press:  01 November 2008

JANINA KOTUS  and
GRZEGORZ ŚWIATEK
JANINA KOTUS
Affiliation:
Faculty of Mathematics and Information Sciences, Warsaw University of Technology, Warsaw 00-661, Poland. e-mail: J.Kotus@impan.gov.pl
GRZEGORZ ŚWIATEK
Affiliation:
Department of Mathematics, Penn State University, University Park, PA 16802, U.S.A. e-mail: swiatek@math.psu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the existence of finite absolutely continuous invariant measures for meromorphic Misiurewicz maps whose Julia set is the whole sphere. In the rational context, these hypotheses imply that such a measure must exist. We show that it is not so for meromorphic maps unless an additional condition on the behavior of the map, which can be stated in terms of its Nevanlinna characteristic, is satisfied.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 3 , November 2008 , pp. 685 - 697
Copyright
Copyright © Cambridge Philosophical Society 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

REFERENCES

[1]Baker, I. N., Kotus, J. and Lü, Y.Iterates of meromorphic functions I. Ergod. Th. Dynam. Sys. 11 (1991), 241–248.CrossRefGoogle Scholar
[2]Baker, I. N., Kotus, J. and Lü, Y.Iterates of meromorphic functions III: Preperiodic domains. Ergod. Th. Dynam. Sys. 11 (1991), 603–618.CrossRefGoogle Scholar
[3]Baker, I. N., Kotus, J. and Lü, Y.Iterates of meromorphic functions IV: Critically finite functions. Results Math. 22 (1992), 651–656.CrossRefGoogle Scholar
[4]Bergweiler, W.Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151–188.CrossRefGoogle Scholar
[5]Bergweiler, W. and Eremenko, A.On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoamericana 11 (1995), 355–373.CrossRefGoogle Scholar
[6]Bock, H.On the dynamics of entire functions on the Julia set. Result. Math. 30 (1996), 16–20.CrossRefGoogle Scholar
[7]Bock, H.Über das Iterationsverhalten meromorpher Funktionen auf der Juliamenge PhD Thesis (Aachen, 1998).Google Scholar
[8]Gol'dberg, A. A. and Ostrovskij, I. V.Raspredelenie znacenij meromorfnych funkcij (Izdat. Nauka, Moskva, 1970).Google Scholar
[9]Graczyk, J., Kotus, J. and Światek, G.Non-recurent meromorphic functions. Fund. Math. 182 (2004), 269–281.CrossRefGoogle Scholar
[10]Graczyk, J. and Smirnov, S.Non-uniform hyperbolicity in complex dynamics I. Prépublications d'Orsay 36 (2001).Google Scholar
[11]Graczyk, J. and Smirnov, S.Non-uniform hyperbolicity in complex dynamics II. Prépublications d'Orsay 36 (2001).Google Scholar
[12]Grzegorczyk, P., Przytycki, F. and Szlenk, W.On iterations of Misiurewicz's rational maps on the Rieman sphere. Ann. Inst. Henri Poincaré 33 (1990), 431–444.Google Scholar
[13]Hayman, W. K.Meromorphic Functions (Oxford Mathemtical Monographs, 1964).Google Scholar
[14]Kotus, J. and Urbański, M.Existence of invariant measures for transcendental subexpanding functions. Math. Zeit. 243 (2003), 25–36.CrossRefGoogle Scholar
[15]Kotus, J. and Urbański, M.Geometry and ergodic theory of non-recurrent elliptic functions. J. Anal. Math. 93 (2004), 35–102.CrossRefGoogle Scholar
[16]Misiurewicz, M.Absolutely continuous measures for certain maps of an interval. Publ. Math. IHES 53 (1981), 17–51.CrossRefGoogle Scholar
[17]Rees, M.Positive measure sets of ergodic rational maps. Ann. Sci Écore Norm. Sup. 19 (1986), 383–407.CrossRefGoogle Scholar
[18]Rippon, P. J. and Stallard, G. M.Iteration of a class of hyperbolic meromorphic functions. Proc. Amer. Math. Soc. 127 (1999), 3251–3258.CrossRefGoogle Scholar
[19]Teichmüller, O.Eine Umkehrung des zweiten Hauptsatzes der Wertverteilungslehre. Deutsche Math. 2 (1937), 96–107.Google Scholar
[20]Kotus, J. and Światek, G.No finite invariant density for Misiurewicz exponential maps. C. R. Acad. Sci. Paris, Ser. I 346 (2008), 559–562.CrossRefGoogle Scholar