CONFORMAL MEASURES FOR MEROMORPHIC MAPS

In this paper we study relations between the existence of a conformal measure on the Julia set J(f) of a transcendental meromorphic map f and the existence of a zero of the topological pressure function t -> P(f, t) for the map f (with respect to the spherical metric). In particular, we show that if f is hyperbolic and admits a t-conformal measure which is not totally supported on the set of escaping points of f, then P(f,t) = 0. On the other hand, for a wide class of maps f, including arbitrary maps with at most finitely many poles and finite set of singular values as well as hyperbolic maps with at most finitely many poles, if P(f, t) = 0, then there exists a t-conformal measure on J(f). This partially answers a question of Mauldin.