Generalized iterations and primitive divisors

Let (gi)i≥1(gi)i≥1 be a sequence of Chebyshev polynomials, each with degree at least two, and define (fi)i≥1(fi)i≥1 by the following recursion: f1=g1f1=g1, fn=gn∘fn−1fn=gn∘fn−1, for n≥2n≥2. Choose α∈Qα∈Q such that {g1n(α):n≥1} is an infinite set. The main result is as follows: If fn(α)=AnBn is written in lowest terms, then for all but finitely many n>0n>0, the numerator, AnAn, has a primitive divisor; that is, there is a prime p   which divides AnAn but does not divide AiAi for any i