to $x_j$, and if ${\lvertc_j\rvert} <1$, we add the sign $\sigma_j= +1$ to $x_j$. Thus the values of Dirichlet data for each open gap are two copies of $[\alpha_j, 
\beta_j]$ glued at the ends, that is, a circle. The set of Dirichlet data is thus an $\ell$-dimensional torus. It is a fundamental result [@vMoer] that the map from $a$’s and $b$’s to Dirichlet data sets up a one-one correspondence to all $a$’s and $b$’s with a given $\Delta$, that is, the set of $a$’s and $b$’s with a given $\Delta$ is an $\ell$-dimensional torus.

The $m$-function associated to $d\rho$ has a meromorphic continuation to the Riemann surface, ${{\mathcal S}}$, with poles precisely at the points $x_j$ on the principal sheet if $\sigma_j=+1$ and on the bottom sheet if $\sigma_j =-1$. $\rho$ has point mass precisely at those $x_j\in (\beta_j, \alpha_{j+1})$ with $\sigma_j =+1$. It has absolutely continuous support exactly the union of the bands, and has no singular part other than the possible point masses in the gaps.

Finally, in the review, we note that the potential theoretic equilibrium measure $dk$ for the set of bands has several critical properties:

If $k(x) = \int_{\alpha_1}^x dk$, then $${\label}{2.15} 
k(\beta_j) = k(\alpha_{j+1}) 