Equilibrium measures for holomorphic endomorphisms of complex projective spaces

Mariusz Urbański, and Anna Zdunik. "Equilibrium measures for holomorphic endomorphisms of complex projective spaces." Fundamenta Mathematicae 220.1 (2013): 23-69. <http://eudml.org/doc/286463>.

@article{MariuszUrbański2013,
abstract = {Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space $ℙ^\{k\}$, k ≥ 1, and let J be the Julia set of f (the topological support of the unique maximal entropy measure). Then there exists a positive number $κ_\{f\} > 0$ such that if ϕ: J → ℝ is a Hölder continuous function with $sup(ϕ) - inf(ϕ) < κ_\{f\}$, then ϕ admits a unique equilibrium state $μ_\{ϕ\}$ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system $(f,μ_\{ϕ\})$ is K-mixing, whence ergodic. Proving almost periodicity of the corresponding Perron-Frobenius operator is the main technical task of the paper. It requires producing sufficiently many “good” inverse branches and controling the distortion of the Birkhoff sums of the potential ϕ. In the case when the Julia set J does not intersect any periodic irreducible algebraic variety contained in the critical set of f, we have $κ_\{f\} = log d$, where d is the algebraic degree of f.},
author = {Mariusz Urbański, Anna Zdunik},
journal = {Fundamenta Mathematicae},
keywords = {holomorphic dynamics; complex projective space; Julia set; topological pressure; equilibrium state; Hölder continuous potential; Perron-Frobenius operator},
language = {eng},
number = {1},
pages = {23-69},
title = {Equilibrium measures for holomorphic endomorphisms of complex projective spaces},
url = {http://eudml.org/doc/286463},
volume = {220},
year = {2013},
}

TY - JOUR
AU - Mariusz Urbański
AU - Anna Zdunik
TI - Equilibrium measures for holomorphic endomorphisms of complex projective spaces
JO - Fundamenta Mathematicae
PY - 2013
VL - 220
IS - 1
SP - 23
EP - 69
AB - Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space $ℙ^{k}$, k ≥ 1, and let J be the Julia set of f (the topological support of the unique maximal entropy measure). Then there exists a positive number $κ_{f} > 0$ such that if ϕ: J → ℝ is a Hölder continuous function with $sup(ϕ) - inf(ϕ) < κ_{f}$, then ϕ admits a unique equilibrium state $μ_{ϕ}$ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system $(f,μ_{ϕ})$ is K-mixing, whence ergodic. Proving almost periodicity of the corresponding Perron-Frobenius operator is the main technical task of the paper. It requires producing sufficiently many “good” inverse branches and controling the distortion of the Birkhoff sums of the potential ϕ. In the case when the Julia set J does not intersect any periodic irreducible algebraic variety contained in the critical set of f, we have $κ_{f} = log d$, where d is the algebraic degree of f.
LA - eng
KW - holomorphic dynamics; complex projective space; Julia set; topological pressure; equilibrium state; Hölder continuous potential; Perron-Frobenius operator
UR - http://eudml.org/doc/286463
ER -