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

Mariusz Urbański¹

¹ Department of Mathematics University of North Texas P.O. Box 311430 Denton, TX 76203-1430, U.S.A.

Fundamenta Mathematicae, Tome 176 (2003) no. 2, pp. 97-125.

Résumé

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.

Détail

DOI : 10.4064/fm176-2-1

Keywords: critically non recurrent rational functions definitions topological pressure proposed coincide study detail gibbs states corresponding potentials t mathop log nolimits their sigma finite invariant versions particular provide sufficient condition their finiteness determine escape rates critically non recurrent rational functions presence parabolic points establish polynomial rate appropriately modified escape extends corresponding result proven context parabolic rational functions part paper introduce class critically tame generalized polynomial like mappings critically tame critically non recurrent generalized polynomial like mapping lder continuous potential sufficiently large exponent has parabolic points topological pressure satisfies mathop sup sufficiently small delta function mapsto delta delta real analytic