Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle
Polynomials orthogonal with respect to a measure supported on the unit circle are studied by means of some recent developed techniques of topological dynamics and differential-dynamical systems. The polynomials considered satisfy the difference equation \[ \Phi_n(z)=(1-|\alpha_n|^2)^{-1/2}\begin{pmatrix} z & \alpha_n\\ \overline\alpha_nz & 1\end{pmatrix} \Phi_{n-1}(z),\quad n\geq 1,\tag{1} \] where \(\alpha_n=g(s^n(\omega))\) for some \(\omega\in\Omega\), \(|\alpha_n|<1\), \(s:\Omega\to\Omega\) is a bimeasurable bijection, \(g:\Omega\to\{z\in\mathbb{C}:|z|<1\}\) is a measurable function, \((\Omega,\mu)\) is a probability space with Borel measure \(\mu\). It is assumed that the reflection coefficients associated with these polynomials form a stationary stochastic ergodic process so that the techniques can applied to the resulting Atkinson-type system obtained from (1) by using a suspension construction. The rotation number defined herein is related to the integrated density of states. A gap-labelling result is proved. It is also shown that, the well-known Kotani-type theorem can be strengthened when \(\Omega\) is compact.
scientific article; zbMATH DE number 952554