2012-05
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ski, Mariusz Urba
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