Random Iteration of Rational Functions Page: 73
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PROOF. Fix p e X. Since X is connected and has diameter at least 46k, the sets Ss(p, 21k)
and Ss(p, 4~k) must intersect X, say at q and r. Since diam3{p, q, r} = 2K and since
#(So) < 2, the pigeonhole principle implies that either p, q, or r is not in B,(SO, k). But
then this point is in K, and p e BS(K, 4'k).
As a result of this claim, we have the bound
ln(Loj[1]) - ln(L0,[1]) osc,x < 1 ln(Loj[1]) - ln(L0 [1]) osc,K + 471(4/k)
< (4 + eM)2- k
Since this is true for all j > nk, we see that I In(L0I [1]) ln(L02[1]) osc,X tends to zero as
ji and j2 approach infinity jointly. Thus if Po e X, then
( L [] (Lj [1]
In[ - In 0.
LO" [1](po) L 2o , 31,J2
(The function whose oc, X norm is being taken vanishes at Po e X.) Thus we are done. D
Without loss of generality suppose that for all w e Q, the following events are satisfied:
A) Events 4.32 and 4.34
B) T(B) cc B
C) (T),N is non-quasilinear and nonsingular
Note that (B) implies that , C X for all w e GQ, since #(X) > 3.
The limit of the sequence (4.4.4) depends on po E X \ So. Since 5 is strongly measurable
and always nonempty, by the selection theorem [[18] Theorem 2.13, p.32] we may choose a
(measurable) random point po E Jo C X \ So.
Fix w e Q. The backwards invariance of X implies that (2.4.1) defines a family of maps
L" : C(X) - C(X). We do not distinguish notationally from this family and from the
original family L" : C(C) - C(C) defined in Section 2.4. We make the following definitions,
whose validity is justified by Events 4.32 and 4.34:
(4.4.5) go : li Lo 1] E C(X) go > 0
4-oL% [1](po) CXg
(4.4.6) A := L" [go] (p,) > 073
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Simmons, David. Random Iteration of Rational Functions, dissertation, May 2012; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc115157/m1/78/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .