Positive Lyapunov Exponents for Quasiperiodic Szego cocycles

TL;DR

This paper derives formulas for Lyapunov exponents of quasiperiodic Szego cocycles, constructs examples with positive exponents, and estimates measure support, advancing understanding of hyperbolicity in quasiperiodic systems.

Contribution

It provides a new formula for averaged Lyapunov exponents and constructs analytic cocycles with uniformly positive exponents, extending previous results to Szego and Schrödinger cocycles.

Findings

Derived a formula for averaged Lyapunov exponents for ergodic Szego cocycles.

Constructed analytic quasiperiodic Szego cocycles with positive Lyapunov exponents.

Estimated Lebesgue measure of support for measures associated with quasiperiodic Verblunsky coefficients.

Abstract

In this paper we first obtain a formula of averaged Lyapunov exponents for ergodic Szego cocycles via the Herman-Avila-Bochi formula. Then using acceleration, we construct a class of analytic quasi-periodic Szego cocycles with uniformly positive Lyapunov exponents. Finally, a simple application of the main theorem in [Y] allows us to estimate the Lebesgue measure of support of the measure associated to certain class of C1 quasiperiodic 2- sided Verblunsky coefficients. Using the same method, we also recover the [S-S] results for Schrodinger cocycles with nonconstant real analytic potentials and obtain some nonuniform hyperbolicity results for arbitrarily fixed Brjuno frequency and for certain C1 potentials.