Random Iteration of Rational Functions Page: 50
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For each r e (0, T) and for each x e r, (z), we have x e (i-1(T(z)). Thus
72: U f{r/}x Tl(z) -(i-1(T(z)).
rie-,1(i T)
(Here F2 is projection onto the second coordinate.) We will be done if 72 is injective.
Suppose that 72(rh, x1) 72(T2, X2), i.e. ZX := X1 X2. Since Q is locally injective
and ( = T o i, T is injective in a neighborhood of 1i(x) = z. Thus T is invertible in a
neighborhood of z, so we have 11 = T-1 o j = 2 in a neighborhood of x. By the identity
principle, j1 =h=2. Thus 72 is injective, and we have shown (4.2.1).
Next, we show (4.2.2). For each i = 1, . . . , rn fix Zxi E UEi at which the maximum in (4.2.2)
is attained. Fix i 1, ..., m, and suppose that z e T-1((i (xi)). If (i(Uj) does not contain
a branch point of T, then by the homotopy lifting principle (i has a unique inverse branch
iz : Ui - C such that rii,z(xi) = z. If furthermore rTi,z is c-good, then z is not counted in
(4.2.2). Thus for each i = 1,..., m and for each z e T- (((xi)), exactly one of the following
the three possibilities holds:
A) (i(Ui) contains a branch point of T.
B) (i(Ui) does not contain a branch point of T, but the inverse branch T/i,z is not c-good
i.e. As(Ti,z (U)) > c.
C) (i(Ui) does not contain a branch point of T, and the inverse branch T/i,z is c-good
i.e. As(Ti,z (U)) < c.
We have already established that category (C) is not counted in (4.2.2). Thus to complete
the proof, it suffices to show that category (A) represents at most 2r deg2(T) pairs (i, z)
(counting multiplicity), and that category (B) represents at most r pairs (i, z) (multiplicity
is not needed since every ramification point is in category (A))
A) It suffices to show that for at most 2r deg(T) values of i 1,..., n, (Q(UI) contains
a branch point. By the Riemann-Hurwitz formula there are at most 2 deg(T) 2
branch points (exactly that many counting multiplicity), and since mult(0) = r,
each branch point is contained in at most r sets of the form ((Ui).50
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Simmons, David. Random Iteration of Rational Functions, dissertation, May 2012; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc115157/m1/55/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .