#### Date of Award

Spring 1-1-2013

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Su-Ion Ih

#### Second Advisor

Katherine Stange

#### Third Advisor

Robert Tubbs

#### Fourth Advisor

Eric Stade

#### Fifth Advisor

Juan Restrepo

#### Abstract

Let (*g _{i}*)

_{i}_{ ≥1}be a sequence of Chebyshev polynomials, each with degree at least two, and define (

*f*)

_{i}

_{i}_{ ≥1}by the following recursion:

*f*

_{1}=

*g*

_{1},

*f*=

_{n}*g*∘

_{n}*f*

_{n}_{–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*(α) = [special characters omitted] is written in lowest terms, then for all but finitely many

_{ n}*n*> 0, the numerator,

*A*, has a primitive divisor; that is, there is a prime

_{n}*p*which divides

*A*but does not divide

_{n}*A*for any

_{i}*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*) =

*g*(

_{n}*f*

_{n}_{–1}(

*x*)) with

*g*coming from [

_{i}*f*], then there is a unique canonical height funtion

*ĥ*

**:**

_{f}*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 *g _{i}* be a sequence of Chebyshev polynomials. Let

*f*given by the following recurrence

*f*

_{ 1}(

*z*) =

*g*

_{1}(

*z*), and

*f*=

_{i}*g*(

_{i}*f*

_{i}_{–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*(α + γ) − γ = [special characters omitted] in lowest terms. Then limn→∞ logAn logBn =1. Finally, some areas of future research are discussed.

_{ n}#### Recommended Citation

Wakefield, Nathan Paul, "Primitive Divisors in Generalized Iterations of Chebyshev Polynomials" (2013). *Mathematics Graduate Theses & Dissertations*. 25.

http://scholar.colorado.edu/math_gradetds/25