Random Iteration of Rational Functions Page: 83
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LEMMA 5.3. Suppose that c/ C 4 \ 1 is a finite set, and suppose F C C is finite with
& (F) C F. Also suppose that (i) of Theorem 4.9 holds. If B C C is a neighborhood of F,
then there exist b e N, 63 > 0, and '2 a neighborhood of a and so that for all T e 2 and
for all pe C \ B,
(5.1.2) diam(T -(p)) > 63*
PROOF. Let r e N be as in Lemma 5.2. Let b be large enough so that 2b > r. Then (5.1.1)
yields that for all T e b and for all x E C \ T-1(F), we have
multT(x) < 2b < deg(T).
Thus no point of C \ T- (F) is totally ramified, and so no point of C \ F is totally branched.
Now, the map (T, p) -+ diam(T-1(p)) is continuous; since C \ B is compact the map
T- inf diam(T-l(p))
peC\B
is continuous, and strictly positive on d/b. Thus there exists 63 > 0 and a neighborhood Mb
of db on which (5.1.2) holds for all p e C \ B; letting '2 be a neighborhood of a such that
SC - b yields the lemma. D
In the following lemma, the idea is to construct a set U which can be used as a domain
on which to bound the Perron-Frobenius operator by considering inverse branches. The set
U should be simply connected and should not contain any branch points of T, so that it has
deg(T) conformally isomorphic preimages. Since U must not contain branch points, it must
vary depending on T, but in a predictable way. Specifically, if S is a perturbation of T then
the the isomorphic preimages of Us under S will be close in some sense to the isomorphic
preimages of UT under T. We now state our lemma:
LEMMA 5.4. Suppose F C C is finite, and suppose B C C is a neighborhood of F. Let
X C \ B. Fixr e N and d > 1. Suppose T E Sod is such that multT(p) < r for all
p e C \ T- (F). Then there exist C < oc, o- > 0, C Md a neighborhood of T, and (()83
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Simmons, David. Random Iteration of Rational Functions, dissertation, May 2012; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc115157/m1/88/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .