in ${{\mathbb{C}}}\backslash\text{bands}$.

For OPUC, the continuous part of the measure, $d\mu$, is supported on $p$ disjoint intervals $\{e^{i\theta}\mid x_j \leq \theta \leq y_j\}$, $j=1, \dots, p$, in $\partial{{\mathbb{D}}}$ with $p$ gaps in between $\{e^{i\theta}\mid y_j\leq \theta \leq x_{j+1}\}$ with $x_{p+1}\equiv 2\pi + x_1$. Each gap has zero or one mass point. The Carathéodory function of the measure $d\mu$, $${\label}{1.10} 
F(z) =\int {\frac}{e^{i\theta}+z}{e^{i\theta}-z}\, d\mu(\theta)$$ has a meromorphic continuation from ${{\mathbb{D}}}$ to the genus $p-1$ hyperelliptic Riemann surface, ${{\mathcal S}}$, associated to $[\prod_{j=1}^p (z-e^{ix_j})(z-e^{iy_j})]^{1/2}$. The surface has a natural projection $\pi:{{\mathcal S}}\to{{\mathbb{C}}}$, and the closure of each gap has a circle as the inverse image. $F$ has a single pole in each such circle, so $p$ in all at $\gamma_1, \dots, \gamma_p$.

Again, the density of zeros is the equilibrium measure for the bands and each band has mass $1/p$ in this measure. See [@OPUC2], especially Chapter 11, for a discussion of periodic OPUC. Our main results for OPUC are:

We can describe the zeros of $\varphi_{np}^* -\varphi_{np}$ exactly (note, not zeros of $\varphi_{np}$).

Asymptotically, as $n\to\infty$, the number of zeros of $\varphi_n$ near each band, $N^{(n,j)}$, obeys $\sup_n {\lvert{\frac}{n}{p} - N^{(n,j)}\rvert}<\infty$, and the points on the bands closest to the zeros 