Artur Avila

A KAM SCHEME FOR SL(2, R) COCYCLES WITH LIOUVILLEAN FREQUENCIES

We develop a new KAM scheme that applies to SL(2,

Complex one-frequency cocycles

{{\mathbb R}}

On the Spectrum and Lyapunov Exponent of Limit Periodic Schrödinger Operators

) cocycles with one frequency, irrespective of any Diophantine condition on the base dynamics. It gives a generalization of Dinaburg–Sinai’s theorem to arbitrary frequencies: under a closeness to constant assumption, the non-Abelian part of the classical reducibility problem can always be solved for a positive measure set of parameters.

Siegel disks with smooth boundaries

We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twenty-first century.

Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts

We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other hand, this last property provides a good control of the periodic approximations of a cocycle, allowing us to show that domination can be characterized, in the presence of a gap in the Lyapunov spectrum, by additional regularity of the dependence of sums of Lyapunov exponents.

Hausdorff dimension and conformal measures of Feigenbaum Julia sets

We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub, both concerning Lyapunov exponents. Indeed, we show that equality holds in Herman’s result. Finally, we give a result about the growth of the spectral radius of products.

Generic singular spectrum for ergodic Schrödinger operators

We exhibit a dense set of limit periodic potentials for which the corresponding one-dimensional Schrödinger operator has a positive Lyapunov exponent for all energies and a spectrum of zero Lebesgue measure. No example with those properties was previously known, even in the larger class of ergodic potentials. We also conclude that the generic limit periodic potential has a spectrum of zero Lebesgue measure.

Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows

We show the existence of angles α ∈ R/Z such that the quadratic polynomial Pα(z) = e2iπαz + z2 has a Siegel disk with C∞-smooth boundary. This result was first announced by R. Perez-Marco in 1993.