Unbounded domains of normality

Abstract

Abstract. Let f be a transcendental entire function of order less than 1/2. We introduce the method of "self-sustaining spread" to study the components of the set of normality of such a function. We give a new proof of the fact that any preperiodic or periodic component of the set of normality of f is bounded. We obtain the same conclusion for a wandering domain if the growth rate of f is never too small.

Uniformly bounded components of normality

“…Let k n (n ∈ N) denote any increasing sequence of positive integers, r 1 > 2 and C > 0 such that 0 < C < 1/4e 2 . Suppose that n 0 is a positive integer so that 2 n 0 −1 C > 2r k 1 1 . The sequence of positive numbers {r n } ∞ n=1 is given inductively by the equation:…”

“…Then | f (w 1 )| R 2 and | f (w 2 2 )| > R c (3) 3 from the conditions of this Lemma. Hence f ( ) must contain an arc joining a point w 2 ∈ C 2 to a point w 1 3 ∈ C 1 3 . This process can be repeated, and f k (U ) contains an arc of f k ( ) which joins a point w k+1 ∈ C k+1 to a point w 1 k+2 ∈ C 1 k+2 .…”

“…Now by a result of [8], there exists a positive constant B such that | f k (w)| | f k (z)| B for all w, z ∈ and for all k n 1 . Now for any k n 1 , we pick points z k , z 1 k ∈ such that f k (z k ) = w k+1 and f k (z 1 k ) = w 1 k+2 . Thus, for each k > n 1 , we have…”

“…Let f (z) be a transcendental entire function. If f satisfies the following condition: there exist two positive constants ε 1 and ε 2 with 0 < ε 1 , ε 2 < 1 such that log dens E( f ) ε 1 (1 •1) where E( f ) = {r > 1 : log m(r, f ) ε 2 log M(r, f )}, (1•2) then we say that f ∈ . contains many classes of transcendental entire functions of any order; see Examples 1, 2, 3 and 4 of the next section.…”

“…In the complex dynamical system theory, it is interesting to investigate the boundedness of all Fatou components of a function f . There is a vast extensive literature on the boundedness of Fatou components, for example see [1,4,12,[17][18][19][20][21]26] for the Fatou sets of transcendental entire functions, [22][23][24]26] for the case of transcendental meromorphic functions. In the above mentioned papers, the authors only considered whether or not all Fatou components of an entire or meromorphic transcendental function are bounded.…”