Random Iteration of Rational Functions Page: I
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Simmons, David. Random Iteration of Rational Functions. Doctor of Philosophy
(Mathematics), May 2012, 143 pp., bibliography, 23 titles.
It is a theorem of Denker and Urbaniski that if T:C-C is a rational map of degree
at least two and if :C--R is Halder continuous and satisfies the "thermodynamic
expanding" condition P(T,) > sup(4), then there exists exactly one equilibrium state p
for T and , and furthermore (C,T,i) 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
Bogenschiitz. Specifically, if (T,Q,P,6) is a holomorphic random dynamical system on C
and Q-- J-k(C) is a Halder continuous random potential function satisfying one of
several sets of technical but reasonable hypotheses, then there exists a unique
equilibrium state of (X,T,4) over (Q,P,6).
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Simmons, David. Random Iteration of Rational Functions, dissertation, May 2012; Denton, Texas. (digital.library.unt.edu/ark:/67531/metadc115157/m1/2/: accessed April 25, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; .