Random Iteration of Rational Functions Page: 64
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Ln2 [f ] Ln, [ f]
(4.3.17)0 <0f]
Ln2[g] osc,K2 Ln,[g] osc,K1
We also rewrite (4.3.10); to complete the proof it suffices to show
(4.3.18)L<([f < (1 -e2)7(/2).
L2[g] osc,K2
Fix p, q e K2. Since Event 4.25 is satisfied for O2w, we have that there exist a disjoint
collection (Ui)72 of open disks, and relatively compact subsets V CC Us satisfying (4.3.2)
and (4.3.3); rewriting (4.3.3) in terms of aw yields
(4.3.19) T"2 ( ) D K2.
For each i= 1,..., m let ( : U - C be the identity map. By (4.3.19), for each i= 1,..., m
there exist xi, ye V with Tn2 (((xi)) =p and T"2(( (yi)) q.
We verify the hypotheses of Proposition 4.23:
* nimeN
* {(1,..., (m} has multiplicity one (since the Uis are disjoint)
* (Ty)n o is a finite sequence of rational maps
* (0j)= is a finite sequence of potential functions
*K1CC
* (4.1.9) - (4. .11) hold for all j 0,... , ni -1
* For each i = 1,...,m, xi i E i
* For each i = 1,...M, (i is injective; in particular, it is u-locally injective since
* 7 ( . C)
* (4.3.2) implies (4.2.15)
* (4.3.15) implies (4.2.1b)
* f, ge C(C) with g>0
* (4.3.14) together with the fact that In(g) osc,K0 C6 implies ln(g) osc,T@ (K) < C6
Thus by Proposition 4.23, we have (4.2.17) for n- ni and K= K1.
Having discharged the quantifiers, we move on to the calculation:64
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Simmons, David. Random Iteration of Rational Functions, dissertation, May 2012; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc115157/m1/69/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .