Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle
type: Item
instance of: scholarly article
title: Rotation number associated with difference equations satisfied by polynomials orthogonal on the unit circle
zbMATH Open document ID: 0863.39003
published in: Journal of Differential Equations
publication date: 1996-12-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 952554
zbMATH Keywords: dichotomy
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: topological dynamics
zbMATH Keywords: differential-dynamical systems
zbMATH Keywords: difference equation
zbMATH Keywords: stationary stochastic ergodic process
zbMATH Keywords: suspension construction
zbMATH Keywords: rotation number
zbMATH Keywords: gap-labelling
author: Jeffrey S. Geronimo
author: Russell Johnson
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0175
OpenAlex ID: W1987279582
DOI: 10.1006/JDEQ.1996.0175
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description: scientific article; zbMATH DE number 952554