the proof of Theorem \[T2.4\] shows that if $z_0$ is not in the bands and is a limit of zeros of $p_{mp+b-1}(z)$, then $p_{mp+b-1}
(z_0)$ goes to zero exponentially (as $\Gamma_-^m$). If this is true for each $b$, then $\sum_{n=0}^\infty {\lvertp_n(z)\rvert}^2 <\infty$, which means $z_0$ is in the pure point spectrum of $d\mu$. Since the bands are also in the spectrum, we have

[P2.6]{} $z_0\in{{\mathbb{C}}}$ is a limit of zeros of $p_n(z)$ [[(]{}]{}all $n$[[)]{}]{} if and only if $z_0\in{\text{\rm{supp}}}(d\mu)$.

[*Remark.*]{} This also follows from a result of Denisov-Simon [@DS2], but their argument, which applies more generally, is more subtle.

OPUC With Periodic Verblunsky Coefficients
==========================================

[s3]{} In this section, we analyze the zeros of OPUC with Verblunsky coefficients obeying . We begin with a summary of the transfer matrices, discriminants, and Abelian functions in this situation. These ideas, while an obvious analog of the OPRL situation, seem not to have been studied before their appearance in [@OPUC2], which is the reference for more details. Many of the consequences of these ideas were found earlier in work of Peherstorfer and Steinbauer [@PS6; @PS1; @PS2; @PS3; @PS4; @PS2000; @PS00].

Throughout, we will suppose that $p$ is even. If $(\alpha_0, \dots, \alpha_{p-1}, 
\alpha_p, 