measures $dk$ for the bands have several critical properties:

If $k(e^{i\theta_0}) = k(\{e^{i\theta}\mid x_1 < \theta < \theta_0\})$, then $${\label}{3.16} 
k(e^{iy_j}) = k(e^{ix_{j+1}}) = {\frac}{j}{p}$$

The Thouless formula holds: $${\label}{3.17} 
\gamma(z) = \int \log {\lvertz-e^{i\theta}\rvert}\, dk(e^{i\theta}) + \log C_B$$ where $\gamma$ is given by and $C_B$ is the capacity of the bands.

We have $${\label}{3.18} 
C_B =\prod_{j=0}^{p-1} (1-{\lvert\alpha_j\rvert}^2)^{1/2}$$

$${\label}{3.19} 
\Gamma_+ (z) = C_B z^{-p/2} \exp \biggl( p\int \log (z-e^{i\theta})\, dk (e^{i\theta})\biggr)$$

This completes the review of periodic OPUC. The analog of Theorem \[T2.1\] does not involve $\Phi_n$ but $\Phi_n - \Phi_n^*$:

[T3.1]{} The zeros of $\Phi_{mp}(z) - \Phi_{mp}^*(z)$ are at the following points:

the $p$ Dirichlet data $z_j$’s in each gap of the period $p$ problem.

the $(m-1)p$ points where $${\label}{3.21} 
k(e^{i\theta}) = {\frac}{k-1}{p} + {\frac}{q}{mp}$$ $k=1, \dots, p$; $q=1, \dots, m-1$.

As noted (and proven several ways in [@OPUC2 Chapter 11]), for a period $mp$ problem, $\Phi_{mp}-\Phi_{mp}^*$ has its zeros, one in each gap. The gaps of the $mp$ problem are the gaps of the original problem plus a closed gap at each point where holds. There is a zero in each closed gap and at each point where holds since then $T_{mp}(z) \binom{1}{1} = c_j^m \binom{1}{1}$.

We now 