Bowen's formula for meromorphic functions

Abstract

Let $f$ be an arbitrary transcendental entire or meromorphic function in the class $\mathcal S$ (i.e. with finitely many singularities). We show that the topological pressure $P(f,t)$ for $t > 0$ can be defined as the common value of the pressures $P(f,t, z)$ for all $z \in \mathbb C$ up to a set of Hausdorff dimension zero. Moreover, we prove that $P(f,t)$ equals the supremum of the pressures of $f|_X$ over all invariant hyperbolic subsets $X$ of the Julia set, and we prove Bowen's formula for $f$, i.e. we show that the Hausdorff dimension of the radial Julia set of $f$ is equal to the infimum of the set of $t$, for which $P(f,t)$ is non-positive. Similar results hold for (non-exceptional) transcendental entire or meromorphic functions $f$ in the class $\mathcal B$ (i.e. with bounded set of singularities), for which the closure of the post-singular set does not contain the Julia set.