Random Iteration of Rational Functions Page: 41
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CHAPTER 4
CONTRACTING NATURE OF THE PERRON-FROBENIUS OPERATOR
4.1. Equicontinuous Perron-Frobenius Operators are Contracting
DEFINITION 4.1. Fix a holomorphic random dynamical system (T, Q, P, 0) on C, and a
random potential function 6 Q - C(C). We say that X C C has the bounded distortion
property if
A) X is closed, connected, contains at least three points, and its complement B C\X
satisfies
T(B) cc B
almost surely.
B) There exists M < c00 so that for all j e N,
(4.1.1) ln(LJ[1]) os,x < M
almost surely. Equivalently, for all n, j e Z with j < n,
(4.1.2)1 lIn(L [1]) osc,x < M
almost surely.
X has the equicontinuity property if X satisfies (A) and if
C) There exists 7 a modulus of continuity such that for all NN
(4.13) p(n) <
Pln(Ln [1]) -
almost surely. Equivalently, for all n, j e Z with j < n,
(4.1.4) p(n) <
a m t sPln(L[1])
almost surely.41
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Simmons, David. Random Iteration of Rational Functions, dissertation, May 2012; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc115157/m1/46/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .