Abstract
In this paper, we first obtain a formula of averaged Lyapunov exponents for ergodic Szegő cocycles via the Herman–Avila–Bochi formula. Then using acceleration, we construct a class of analytic quasiperiodic Szegő cocycles with uniformly positive Lyapunov exponents. Finally, a simple application of the main theorem in Young (1997 Ergod. Theory Dyn. Syst. 25 483–504) allows us to estimate the Lebesgue measure of support of the measure associated with certain class of C1 quasiperiodic 2-sided Verblunsky coefficients. Using the same method, we also recover the Sorets and Spencer (1991 Commun. Math. Phys. 142 543–66) results for Schrödinger cocycles with nonconstant real analytic potentials and obtain some nonuniform hyperbolicity results for arbitrarily fixed Brjuno frequency and for certain C1 potentials.
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