Iterations of rational functions and the distribution of the values of the Poincar� functions

Abstract

Let R n be the nth iterate of the function R. In the classical works of Julia, Fatou, and Latt6s one has pointed out the close connection between the distribution of the values of the function f and the distribution of the roots of the equation Rn(z) = a. We shall continue the investigation of this connection, making use of the Nevanlinna theory of the distribution of the values of meromorphic functions [2, 3]. In particular, we shall give a new proof of the uniqueness of an invariant balanced measure of the function R and of the asymptotically uniform distribution of the roots of the equation Rn(z) = a with respect to this measure [4, 5]. The definition of a balanced measure and the precise formulation of the result are given in Sec. 5. All the facts regarding iterates of rational functions, used in this paper, can be found in [6, Chap. VIII]. 1. Exceptional Values. By definition, the set E(R) of the exceptional values of a rational function R consists of those a E ~, such that the equation Rn(z) = a, n E N, have in totality a finite number of roots. In other words, the points a E E(R) have only a finite number of antecedents. As it is known, card E(R) _< 2. The rational function R and S are said to be conjugate if R o ~o = ~o o S for some linear fractional function ~o. If card E(R) = 2, then the function R is conjugate with z ~ z -+d. If card E(R) = 1, then R is conjugate with a polynomial of degree d. We denote by Epq) = {aE if: : f (z) ~ a, z E ~} the set of Picard exceptional values of the function f. If f is the Poincar6 function for R, then Ep(f) = E(R). In particular, f is entire if and only if R is a polynomial. We need one elementary lemma. LEMMA 1. If the equation R3(z) = a has a root of order d 3, then a E E(R). For the sake of completeness, we give the proof of this lemma. Assume that the equation R3(z) = a has a root of order d 3, Then it has only one root. In this case the equation R(z) = a has a unique root a_ 1 of order d and the equation R(z) = a_ 1 has a unique root a_ 2 of order d. Since also the equation R(z) = a_ 2 has a unique root d, we conclude that among the points a, a_l , a_ 2 at least two are equal (since the total number of critical points of the function R, taking into account multiplicities, is equal to 2d 2). From here it follows that a E E(R). 2. The Nevanlinna Characteristics [2, 3]. For an arbitrary function f, meromorphic in C, we set