Dynamics in the Szeg\"o class and polynomial asymptotics
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, we show that to each J in Sz(E) there is a unique element J' in the isospectral torus, TE, so that the left-shifts of J are asymptotic to the orbit J'm on TE. 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.
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