It is a theorem of Denker and Urbański that if T:ℂ→ℂ is a rational map of degree at least two and if ϕ:ℂ→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 (ℂ,T,μ) is metrically exact. We extend these results to the case of a holomorphic random dynamical system on ℂ, using the concepts of relative pressure and relative entropy of such a system, and the variational principle of Bogenschütz. Specifically, if (T,Ω,P,θ) is a holomorphic random dynamical system on ℂ and ϕ:Ω→ ℋα(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 (X,P,ϕ) over (Ω,Ρ,θ).
Identifer | oai:union.ndltd.org:unt.edu/info:ark/67531/metadc115157 |
Date | 05 1900 |
Creators | Simmons, David |
Contributors | Urbański, Mariusz, Cherry, William, 1966-, Fishman, Lior |
Publisher | University of North Texas |
Source Sets | University of North Texas |
Language | English |
Detected Language | English |
Type | Thesis or Dissertation |
Format | Text |
Rights | Public, Simmons, David, Copyright, Copyright is held by the author, unless otherwise noted. All rights Reserved. |
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