Entire functions with bounded Fatou components

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Starting with the work of I. N. Baker that appeared in 1981, many authors have studied the question of under what circumstances every component of the Fatou set of a transcendental entire function must be bounded. In particular, such functions have no domains now known as Baker domains, and no completely invariant domains. There may be wandering domains but not the familiar and more easily constructed un-bounded ones that often appear for functions defined by simple explicit formulas.

Two types of criteria are involved in the partial answers obtained for this question: the order of growth, and the regularity of the growth of the function.

Baker himself showed that a function of sufficiently slow growth has only bounded Fatou components and noted that order 1/2 minimal type is the best condition one could hope for. Subsequently his results have been extended to order less than 1/2 except for wandering domains, and also to wandering domains if the growth satisfies, in addition, a mild regularity condition. Similar results have been obtained also for certain functions of faster growth provided that the growth is sufficiently regular.

In this paper we review the results achieved and the methods involved in this area.
Original languageEnglish (US)
Title of host publicationTranscendental dynamics and complex analysis
PublisherCambridge University Press
Pages187-216
Number of pages30
Volume348
DOIs
StatePublished - 2008

Publication series

NameLondon Math. Soc. Lecture Note Ser.
PublisherCambridge Univ. Press, Cambridge

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