On Symmetries of the Feinberg-Zee Random Hopping Matrix

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 $mgeq 2$, 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 $pin S$, including those of $P_m$, this partially answering a conjecture of the second author.