Conformal measures for meromorphic maps

TL;DR

This paper explores the relationship between conformal measures and the topological pressure function for transcendental meromorphic maps, establishing conditions under which conformal measures exist and linking them to the pressure's zeros.

Contribution

It provides new results connecting conformal measures and the pressure function for a broad class of meromorphic maps, partially answering Mauldin's question.

Findings

If $f$ is hyperbolic and a $t$-conformal measure exists not fully supported on escaping points, then $P(f, t) = 0.

For many meromorphic maps, including those with finitely many poles and singular values, $P(f, t) = 0$ implies the existence of a $t$-conformal measure.

The results extend understanding of conformal measures in complex dynamics, especially for transcendental 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.