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A finiteness theorem for a dynamical class of entire functions

Published online by Cambridge University Press:  19 September 2008

Lisa R. Goldberg  and
Linda Keen
Lisa R. Goldberg
Affiliation:
C.U.N.Y., 33 West 42 Street, New York, NY 10036, USA
Linda Keen
Affiliation:
C.U.N.Y., 33 West 42 Street, New York, NY 10036, USA
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Abstract

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We define a class Σ of entire functions whose covering properties are similar to those of rational maps. The set Σ is closed under composition of functions, and we show that when regarded as dynamical systems of the plane, the elements of Σ share many properties with rational maps. In particular, they have finite dimensional spaces of quasiconformal deformations, and they contain no wandering domains in their stable sets.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 2 , June 1986 , pp. 183 - 192
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[1]Ahlfors, L. V.. Lectures on Quasiconformal Mappings. Van Nostrand, 1966.Google Scholar
[2]Ahlfors, L. V. & Bers, L.. Riemann's mapping theorem for variable metrics. Ann. of Math. 72 (1960), 385–404.CrossRefGoogle Scholar
[3]Baker, I. N.. The domains of normality of an entire function. Ann. Acad. Sci. Fenn. Ser. A.I. Math. 1 (1975), 277–283.CrossRefGoogle Scholar
[4]Baker, I. N.. Wandering domains in the iteration of entire functions. Preprint, 1983.Google Scholar
[5]Bers, L.. Fiber spaces over Teichmüller spaces. Acta Math. 130 (1973), 89–126.CrossRefGoogle Scholar
[6]Bers, L.. On Sullivan's proof of the finiteness theorem and the eventual periodicity theorem. Preprint.Google Scholar
[7]Fatou, P.. Sur l'itération des fonctions transcendentes entières. Acta math. 47 (1920), 337–370.CrossRefGoogle Scholar
[8]Fricke, R. & Klein, F.. Vorlesungen über die Theorie der Autorhorphen Funktionen, Vol. 1, part II.2; reprinted 1965, Academic Press.Google Scholar
[9]Goldberg, L. & Keen, L.. On stable regions for a class of entire functions. In preparation.Google Scholar
[10]Nevanlinna, R.. Analytic Fundions. Springer Verlag, 1970.CrossRefGoogle Scholar
[11]Nielsen, J.. Untersuchung zur Topologie der geschlossen zweiseitigen Flächen. Ada math. 50 1927.Google Scholar
[12]Seigel, C. L.. Bermerkung zu einem Satz von Jakob Nielsen. In Gesammelte Abhandlungen. Springer-Verlag, 1966, Vol. 3, pp. 92, 96.CrossRefGoogle Scholar
[13]Sullivan, D.. Quasiconformal homeomorphisms and dynamics I. Preprint.Google Scholar
[14]Tangerman, F.. Personal communication.Google Scholar