{\overline}{\varphi_n (1/\bar z)}$$ and $${\label}{1.4} 
\rho_n =(1-{\lvert\alpha_n\rvert}^2)^{1/2}$$

In this paper, we focus on the case where the Jacobi coefficients $\{a_n\}_{n=1}^\infty, 
\{b_n\}_{n=1}^\infty$ or the Verblunsky coefficients $\{\alpha_n\}_{n=0}^\infty$ are periodic, that is, for some $p$, $${\label}{1.5} 
a_{n+p} =a_n \qquad b_{n+p} =b_n$$ or $${\label}{1.6} 
\alpha_{n+p}=\alpha_n$$ It should be possible to say something about perturbations of a periodic sequence, say $\alpha_n^{(0)}$, which obeys and $\alpha_n = \alpha_n^{(0)} + 
\delta\alpha_n$ with ${\lvert\delta\alpha_n\rvert}\to 0$ sufficiently fast. We leave the details to be worked out elsewhere.

To describe our results, we begin by summarizing some of the basics of the structure of the measures and recursion relations when or holds. We will say more about this underlying structure in the sections below. In this introduction, we will assume that all gaps are open, although we don’t need and won’t use that assumption in the detailed discussion.

When holds, the continuous part of the underlying measure, $d\rho$, on ${{\mathbb{R}}}$ is supported on $p$ closed intervals $[\alpha_j, \beta_j]$, $j=1, \dots, p$, called bands, with gaps $(\beta_j, \alpha_{j+1})$ in between. Each gap has zero or one mass point. The $m$-function of the measure $d\rho$, $${\label}{1.7} 
m(z) = \int {\frac}{d\rho(x)}{x-z}$$ has a meromorphic continuation to the genus $p-1$ hyperelliptic Riemann surface, 