Primitive Divisors in Generalized Iterations of Chebyshev Polynomials

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: f₁ = g₁, f_n = g_n ∘ f_n–1, for n ≥ 2. Choose α ∈ $Q$ such that {$gn1$(α) : n ≥ 1} is an infinite set. The main result is as follows: Let γ ∈ {0, ±1}, if f _n(α) = $AnBn$ 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 = is a generalized iteration of rational maps formed by f _n(x) = g_n( f_n–1(x)) with g_i coming from [f], then there is a unique canonical height funtion ĥ_f : K → $R$ 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) = g₁(z), and f_i = g_i( f_i–1(z)) for i ≥ 2. Pick any α ∈ $Q$ with |α + γ| < 2, such that α + γ is not pre-periodic for one hence any Chebyshev polynomial. Write f _n(α + γ) − γ = $AnBn$ in lowest terms. Then limn→∞ logAn logBn =1. Finally, some areas of future research are discussed.