9 - Entire functions with bounded Fatou components

Summary

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.

INTRODUCTION

Let f be a transcendental entire function. We consider the question, initiated by I.N. Baker in 1981, of under what circumstances all the components of the Fatou set of f are bounded.