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

Mariusz Urbański. "Thermodynamic formalism, topological pressure, and escape rates for critically non-recurrent conformal dynamics." Fundamenta Mathematicae 176.2 (2003): 97-125. <http://eudml.org/doc/283106>.

@article{MariuszUrbański2003,
abstract = {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.},
author = {Mariusz Urbański},
journal = {Fundamenta Mathematicae},
keywords = {non-recurrent rational functions; topological pressure; Gibbs states; escape rates; parabolic rational functions; tame generalized polynomial-like mappings},
language = {eng},
number = {2},
pages = {97-125},
title = {Thermodynamic formalism, topological pressure, and escape rates for critically non-recurrent conformal dynamics},
url = {http://eudml.org/doc/283106},
volume = {176},
year = {2003},
}

TY - JOUR
AU - Mariusz Urbański
TI - Thermodynamic formalism, topological pressure, and escape rates for critically non-recurrent conformal dynamics
JO - Fundamenta Mathematicae
PY - 2003
VL - 176
IS - 2
SP - 97
EP - 125
AB - 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.
LA - eng
KW - non-recurrent rational functions; topological pressure; Gibbs states; escape rates; parabolic rational functions; tame generalized polynomial-like mappings
UR - http://eudml.org/doc/283106
ER -