Graduate Thesis Or Dissertation
Primitive Divisors in Generalized Iterations of Chebyshev Polynomials Public Deposited
 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} = gn ∘ f_{n}_{1}, for n ≥ 2. Choose α ∈ Q such that {g^{n1} (α) : n ≥ 1} is an infinite set. The main result is as follows: Let γ ∈ {0, ±1}, if f n(α) = A_{n}/B_{n} is written in lowest terms, then for all but finitely many n > 0, the numerator, A_{n}, has a primitive divisor; that is, there is a prime p which divides A_{n} but does not divide A_{i} 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 = (f_{i})^{∞}_{i=1} is a generalized iteration of rational maps formed by fn(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 preperiodic 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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 2013
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 20200121
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