Random Iteration of Rational Functions
per
Simmons, David
Major Professor
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ski, Mariusz Urba
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Cherry, William
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Fishman, Lior
www.unt.edu
Denton, Texas
University of North Texas
2012-05
eng
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).
Random dynamics
complex dynamics
thermodynamic formalism
UNTETD
UNT
public
Simmons, David
copyright
Copyright is held by the author, unless otherwise noted. All rights Reserved.
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Department of Mathematics
Mathematics
Doctoral