Conformal measures for meromorphic maps

Abstract

In this paper we study the relation between the existence of a conformal measure on the Julia set $J(f)$ of a transcendental meromorphic map $f$ and the existence of zero of the topological pressure function $t \mapsto P(f, t)$ for the map $f$. In particular, we show that if $f$ is hyperbolic and there exists a $t$-conformal measure which is not totally supported on the set of escaping points, 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 and hyperbolic maps with at most finitely many poles and bounded set of singular values, if $P(f, t) = 0$, we construct a $t$-conformal measure on $J(f)$. This partially answers a question of R.D. Mauldin.