## Mathematics Graduate Theses & Dissertations

Spring 1-1-2013

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Su-Ion Ih

Katherine Stange

Robert Tubbs

Juan Restrepo

#### Abstract

Let (gi)i ≥1 be a sequence of Chebyshev polynomials, each with degree at least two, and define (fi) i ≥1 by the following recursion: f1 = g1, fn = gn fn–1, for n ≥ 2. Choose α ∈ [special characters omitted] such that {[special characters omitted](α) : n ≥ 1} is an infinite set. The main result is as follows: Let γ ∈ {0, ±1}, if f n(α) = [special characters omitted] is written in lowest terms, then for all but finitely many n > 0, the numerator, An, has a primitive divisor; that is, there is a prime p which divides An but does not divide Ai for any i < n.

In addition to the main result, several of the tools developed to prove the main result may be of interest.

A key component of the main result was the development of a generalization of canonical height. Namely: If [f] is a set of rational maps, all commuting with a common function f, and f = [special characters omitted] is a generalized iteration of rational maps formed by f n(x) = gn( fn–1(x)) with gi coming from [f], then there is a unique canonical height funtion ĥf : K → [special characters omitted] which is identical to the canonical height function associated to f.

Another key component of the main result was proving that under certain circumstances, being acted upon by a Chebyshev polynomial does not lead to significant differences between the size of the numerator and denominator of the result. Specifically, let γ ∈ {0, ±1, ±2} be fixed, and gi be a sequence of Chebyshev polynomials. Let f given by the following recurrence f 1(z) = g1(z), and fi = gi( fi–1(z)) for i ≥ 2. Pick any α ∈ [special characters omitted] with |α + γ| < 2, such that α + γ is not pre-periodic for one hence any Chebyshev polynomial. Write f n(α + γ) − γ = [special characters omitted] in lowest terms. Then limn→∞ logAn logBn =1. Finally, some areas of future research are discussed.

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