  In this paper we study the spectrum $\Sigma$ of the infinite Feinberg-Zee
random hopping matrix, a tridiagonal matrix with zeros on the main diagonal and
random $\pm 1$'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 (arXiv:1412.1937, Random Matrices:
Theory Appl.}, {\bf 4} 1550016 (2015)) has shown that the so-called periodic
part $\Sigma_\pi$ of $\Sigma$, conjectured to be the whole of $\Sigma$ and
known to include the unit disk, satisfies $p^{-1}(\Sigma_\pi) \subset
\Sigma_\pi$ 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 $P_m(\lambda) = \lambda U_{m-1}(\lambda/2)$, for $m\geq 2$, where
$U_n(x)$ is the Chebychev polynomial of the second kind of degree $n$. We also
explore implications of these inverse polynomial mappings, for example showing
that $\Sigma_\pi$ is the closure of its interior, and contains the filled Julia
sets of infinitely many $p\in S$, including those of $P_m$, this partially
answering a conjecture of the second author.
