obey an estimate like .

$z\in{{\mathbb{C}}}$ is a limit of zeros of $\varphi_n$ if and only if $z$ lies in ${\text{\rm{supp}}}(d\mu)$.

There are at most $2p+2b-1$ points which are limits of zeros of $\varphi_{mp+b}$ for each $b=1, \dots, p$ and, except for these limits, zeros have no accumulation points in ${{\mathbb{C}}}\backslash\text{bands}$.

In Section \[s2\], we discuss OPRL when holds, and in Section \[s3\], OPUC when holds. Each section begins with a summary of transfer matrix techniques for periodic recursion coefficients (Floquet theory).

While I am unaware of any previous work on the precise subject of Sections \[s2\] and \[s3\], the results are closely related to prior work of Peherstorfer [@Peh03; @Peh04], who discusses zeros in terms of measures supported on a union of bands with a particular structure that overlaps our class of measures. For a discussion of zeros for OPUC with two bands, see [@LukPeh].

These papers also consider situations where the recursion coefficients are only almost periodic. For any finite collection of closed intervals on ${{\mathbb{R}}}$ or closed arcs on $\partial{{\mathbb{D}}}$, there is a natural isospectral torus of OPRL or OPUC where the corresponding $m$- or $F$-function has minimal degree on the Riemann surface (see, e.g., [@OPUC2 Section 