On Symmetries of the Feinberg-Zee Random Hopping Matrix

In this paper we study the spectrum $\mathrm{\Sigma }$ 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 ${\mathrm{\Sigma }}_{\pi }$ of $\mathrm{\Sigma }$, conjectured to be the whole of $\mathrm{\Sigma }$ and known to include the unit disk, satisfies $p^{-1}(\Sigma_\pi) \subset \Sigma_\pi$foraninfiniteclass$S$ofmonicpolynomials$p$. In this paper we make very explicit the membership of $S$, in particular showing that it includes $P_m(\lambda )=\lambda U_{m-1}(\lambda \mathrm{/}2)$, for $m\ge 2$, 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 ${\mathrm{\Sigma }}_{\pi }$ 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.