Existence of Perfect Picard Sets

Abstract

LetEbe a totally disconnected compact set in thez-plane and letΩbe its complement with respect to the extended 2-plane. ThenΩis a domain and we can consider a single-valued meromorphic functionf(z) inΩwhich has a transcendental singularity at each point ζ ∊E. Suppose thatEis a null-set of the classWin the sense of Kametani [4] (= the classNBin the sense of Ahlfors and Beurling [1]).

Perfect Picard Set of Positive Capacity

“…Further we denote by C{w; δ) (δ>0) the spherical disc with center w and with chordal radius δ. For the proof of our theorem we shall need the following lemmas which were given in [1]. In the proof of this lemma it is shown that holds good the following fact, which we state as a lemma.…”

“…For any two points w and w' in the w/-plane we denote by ίw f w'l the chordal distance between them, that is, I""'"'I ^-if ^ oo and ίw 9 w'l = Further we denote by C{w; δ) (δ>0) the spherical disc with center w and with chordal radius δ. For the proof of our theorem we shall need the following lemmas which were given in [1]. Let E be a Cantor set on the closed interval / 0 : C -1/2, 1/2] on the real axis of the z-plane with successive ratios ξ n , 0<ς» = 2\^Λ<2/3.…”

Dynamics of functions meromorphic outside a small set

“…Matsumoto [34] showed that if ξ n+1 = o(ξ 2 n ) as n → ∞ then every f ∈ M(E) is in MP 2 . Matsumoto [34] showed that if ξ n+1 = o(ξ 2 n ) as n → ∞ then every f ∈ M(E) is in MP 2 .…”

Picard sets admitting exceptionally ramified meromorphic functions

“…If any function of M E has at most two Picard exceptional values at each singularity ζ<^E, E is called a Picard set. The existence of perfect Picard sets was shown by means of Cantor sets in Matsumoto [4]. A meromorphic function f(z) of M E is said to be exceptionally ramified at a singularity ζe£, if there exist values w k , l<k<q, and positive integers v k , l^k<q, with such that, in some neighborhood of ζ, the multiplicity of any w k -point of f(z) is not less than v k .…”

“…LEMMA 1 (Carleson [2] and Matsumoto [4]). Let w=g(ξ) be a single-valued regular function in an annulus l<\ξ\<μ 2 omitting two values 0 and 1.…”