open gaps. (In some sense, the OPRL case, where the genus $\ell$ is the number of gaps, has $\ell +1$ gaps also, but one gap is ${{\mathbb{R}}}\backslash 
[\alpha_1, \beta_p]$ which includes infinity.) $\Gamma_-$ is the analytic continuation of $\Gamma_+$ to the second sheet.

The Dirichlet data are partly these points in $\partial{{\mathbb{D}}}$, $z_j$, $${\label}{3.15} 
T_p(z) \binom{1}{1} = c_z \binom{1}{1}$$ It can be shown there is one such $z_j$ in each gap (including closed gaps) for the $p$ roots of $\varphi_p(z) - \varphi_p^*(z)$. We let $c_j = c_{z_j}$. If $z_j$ is at a gap edge, ${\lvertc_j\rvert}=1$; otherwise ${\lvertc_j\rvert}\neq 1$. If ${\lvertc_j\rvert} >1$, we add sign $-1$ to $z_j$ and place the Dirichlet point on the lower sheet of ${{\mathcal S}}$ at point $z_j$. If ${\lvertc_j\rvert}<1$, we add sign $+ 1$ and put the Dirichlet point on the initial sheet. $+1$ points correspond to pure points in $d\mu$.

As in the OPRL case, the set of possible Dirichlet data points is a torus, but now of dimension $\ell +1$. This torus parametrizes those $\mu$ with periodic $\alpha$’s and discriminant $\Delta$.

The $F$-function, , has a meromorphic contribution to ${{\mathcal S}}$ with poles precisely at the Dirichlet data points.

The potential theoretic equilibrium 