Random Iteration of Rational Functions

Description:

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).

Creator(s): Simmons, David
Creation Date: May 2012
Partner(s):
UNT Libraries
Collection(s):
UNT Theses and Dissertations
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Total Uses: 110
Past 30 days: 1
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Publisher Info:
Publisher Name: University of North Texas
Publisher Info: www.unt.edu
Place of Publication: Denton, Texas
Date(s):
  • Creation: May 2012
Description:

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).

Degree:
Discipline: Mathematics
Level: Doctoral
PublicationType: Disse
Language(s):
Subject(s):
Keyword(s): Random dynamics | complex dynamics | thermodynamic formalism
Contributor(s):
Partner:
UNT Libraries
Collection:
UNT Theses and Dissertations
Identifier:
  • ARK: ark:/67531/metadc115157
Resource Type: Thesis or Dissertation
Format: Text
Rights:
Access: Public
Holder: Simmons, David
License: Copyright
Statement: Copyright is held by the author, unless otherwise noted. All rights Reserved.