Orthogonal polynomials on several intervals: accumulation points of recurrence coefficients and of zeros

Let $E = \cup_{j = 1}^l [a_{2j-1},a_{2j}],$ $a_1 < a_2 < ... < a_{2l},$ $l \geq 2$ and set {\boldmath\omega}(\infty) =(\omega_1(\infty),...,\omega_{l-1}(\infty)), where $\omega_j(\infty)$ is the harmonic measure of $[a_{2 j - 1}, a_{2 j}]$ at infinity. Let $\mu$ be a measure which is on $E$ absolutely continuous and satisfies Szeg\H{o}'s-condition and has at most a finite number of point measures outside $E$, and denote by $(P_n)$ and $({\mathcal Q}_n)$ the orthonormal polynomials and their associated Weyl solutions with respect to $d\mu$, satisfying the recurrence relation $\sqrt{\lambda_{2 + n}} y_{1 + n} = (x - \alpha_{1 + n}) y_n -\sqrt{\lambda_{1 + n}} y_{-1 + n}$. We show that the recurrence coefficients have topologically the same convergence behavior as the sequence (n {\boldmath\omega}(\infty))_{n\in \mathbb N} modulo 1; More precisely, putting ({\boldmath\alpha}^{l-1}_{1 + n}, {\boldmath\lambda}^{l-1}_{2 + n}) = $(\alpha_{[\frac{l 1}{2}]+1+n},...,$ $\alpha_{1+n},...,$ $\alpha_{-[\frac{l-2}{2}]+1+n},$ $\lambda_{[\frac{l-2}{2}]+2+n},$ $...,\lambda_{2+n},$ $...,$ $\lambda_{-[\frac{l-1}{2}]+2+n})$ we prove that ({\boldmath\alpha}^{l-1}_{1 + n_\nu}, {\boldmath\lambda}^{l-1}_{2 + n_\nu})_{\nu \in \mathbb N} converges if and only if (n_\nu {\boldmath\omega}(\infty))_{\nu \in \mathbb N} converges modulo 1 and we give an explicit homeomorphism between the sets of accumulation points of ({\boldmath\alpha}^{l-1}_{1 + n}, {\boldmath\lambda}^{l-1}_{2 + n}) and (n{\boldmath\omega}(\infty)) modulo 1.