In this paper we study the spectrum Σ of the infinite Feinberg-Zee random hopping matrix, a tridiagonal matrix with zeros on the main diagonal and random ±1‘s on the first sub- and super-diagonals; the study of this non-selfadjoint random matrix was initiated in Feinberg and Zee (Phys. Rev. E 59 (1999), 6433-6443). Recently Hagger (Random Matrices: Theory Appl., 4 1550016 (2015)) has shown that the so-called periodic part Σπ of Σ, conjectured to be the whole of Σ and known to include the unit disk, satisfies p-1(Σπ) ⊂ Σπ for an infinite class S of monic polynomials p. In this paper we make very explicit the membership of S, in particular showing that it includes Pm(λ) = λUm-1(λ/2), for m ≥ 2, where Un(x) is the Chebychev polynomial of the second kind of degree n. We also explore implications of these inverse polynomial mappings, for example showing that Σπ is the closure of its interior, and contains the filled Julia sets of infinitely many p ∈ S, including those of Pm, this partially answering a conjecture of the second author.
