2013/03/11 by David Simmons, Simmons, David · 1 citation
Computer Science · Mathematics · #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #math.DS
paper · pdf · doi:10.48550/arxiv.1303.2705
arxiv created 2013/03/11 · openalex publication_date 2013/03/11 · arxiv updated 2013/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is a theorem of Denker and Urbański ('91) that if T:\mathbb C→\mathbb C is a rational map of degree at least two and if ϕ:\mathbb C→\mathbb R is Hölder continuous and satisfies the "thermodynamic expanding" condition P(T,ϕ) > sup(ϕ), then there exists exactly one equilibrium state μ for T and ϕ, and furthermore (\mathbb C,T,μ) is metrically exact. We extend these results to the case of a holomorphic random dynamical system on \mathbb C, using the concepts of relative pressure and relative entropy of such a system, and the variational principle of Bogenschütz ('92/'93). Specifically, if (T,Ω,\textbf P,θ) is a holomorphic random dynamical system on \mathbb C and ϕ:Ω→ Hα(\mathbb C) is a Hölder continuous random potential function satisfying one of several sets of technical but reasonable hypotheses, then there exists a unique equilibrium state of (\mathbb X,\mathbb T,ϕ) over (Ω,\textbf P,θ). Also included is a general (non-thermodynamic) discussion of random dynamical systems acting on \mathbb C, generalizing several basic results from the deterministic case.