## Thermodynamic formalism, topological pressure, and escape rates for critically non-recurrent conformal dynamics

### Volume 176 / 2003

#### Abstract

We show that for critically non-recurrent rational functions all the definitions of topological pressure proposed in [12] coincide for all $t\ge 0$. Then we study in detail the Gibbs states corresponding to the potentials $-t\mathop {\rm log}\nolimits |f'|$ and their $\sigma $-finite invariant versions. In particular we provide a sufficient condition for their finiteness. We determine the escape rates of critically non-recurrent rational functions. In the presence of parabolic points we also establish a polynomial rate of appropriately modified escape. This extends the corresponding result from [6] proven in the context of parabolic rational functions. In the last part of the paper we introduce the class of critically tame generalized polynomial-like mappings. We show that if $f$ is a critically tame and critically non-recurrent generalized polynomial-like mapping and $g$ is a Hölder continuous potential (with sufficiently large exponent if $f$ has parabolic points) and the topological pressure satisfies ${\rm P}(g)>\mathop {\rm sup}(g)$, then for sufficiently small $\delta >0$, the function $t\mapsto {\rm P}(tg)$, $t\in (1-\delta ,1+\delta )$, is real-analytic.