Lyapunov exponenets of quasi-periodically driven stystems

The Schrödinger operator with a quasi-periodic potential has attracted very much attention during the last 20-years.

When the potential is real-analytic we have by now a good description of the spectral and dynamical properties of the operator in the regimes of "small" and "large" potentials; in particular in the "one-frequency" case. J. Bourgain has been one of the leaders in this development.

In recent years a global theory for the Schrödinger operator with a "one-frequency" analytic quasi-periodic potential has been introduced by A. Avila.

This theory explains what can happen in the transition region from "small" potential to "large"; but it does not tell us when it happens.

From the point of view of the proofs of (most of) the above results analyticity is a crucial assumption; but from the point of view of the problem itself it is not at all a natural condition. Results outside the analytic regime also exist, for example important contributions by T. Spencer and Ya. Sinai; but these results are so far much more sparse and incomplete.

It has been shown that new phenomena can occure outside the class of analytic potentials, so the picture must be more complicated when reularity is lost.

In this project we want to use dynamical systems methods to (1) show that the case of large "two-frequency" finitely smooth potentials can be treated, and (2) provide new explicit examples in the tranistion region. A key is to control the Lyapunov exponents.