Dynamics in the Szeg\"o class and polynomial asymptotics

TL;DR

This paper explores the dynamics of the Szeg"o class for Parreau-Widom sets, establishing asymptotic behaviors of orthogonal polynomials and their ratios through the isospectral torus and Jost functions.

Contribution

It introduces the Szeg"o class for arbitrary Parreau-Widom sets and characterizes the asymptotic behavior of orthogonal polynomials under the left shift dynamics.

Findings

Unique correspondence between Szeg"o class elements and the isospectral torus.

Existence of a limit for the ratio of orthogonal polynomials expressed via Jost functions.

Asymptotic behavior of orthogonal polynomials described for all elements in the Szeg"o class.

Abstract

We introduce the Szeg\"o class, Sz(E), for an arbitrary Parreau-Widom set E in R and study the dynamics of its elements under the left shift. When the direct Cauchy theorem holds on C\E, we show that to each J in Sz(E) there is a unique element J' in the isospectral torus, T_E, so that the left-shifts of J are asymptotic to the orbit {J'_m} on T_E. Moreover, we show that the ratio of the associated orthogonal polynomials has a limit, expressible in terms of Jost functions, as the degree n tends to infinity. This enables us to describe the large n behaviour of the orthogonal polynomials for every J in the Szeg\"o class.