Abstract/Summary

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 ({\it Phys. Rev. E} {\bf 59} (1999), 6433--6443). Recently Hagger ({\em Random Matrices: Theory Appl.}, {\bf 4} 1550016 (2015)) has shown that the so-called {\em 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 $\cS$ of monic polynomials $p$. In this paper we make very explicit the membership of $\cS$, 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 $p\in \cS$, including those of $P_m$, this partially answering a conjecture of the second author.