We show that for critically non-recurrent rational functions all the definitions of topological pressure proposed in [12] coincide for all t ≥ 0. Then we study in detail the Gibbs states corresponding to the potentials -tlog|f'| and their σ-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 P(g) > sup(g), then for sufficiently small δ >0, the function t↦ P(tg), t ∈ (1-δ,1+δ), is real-analytic.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-2-1,
author = {Mariusz Urba\'nski},
title = {Thermodynamic formalism, topological pressure, and escape rates for critically non-recurrent conformal dynamics},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {97-125},
zbl = {1022.37033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-2-1}
}
Mariusz Urbański. Thermodynamic formalism, topological pressure, and escape rates for critically non-recurrent conformal dynamics. Fundamenta Mathematicae, Tome 177 (2003) pp. 97-125. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm176-2-1/