# Loss of uniform hyperbolicity and regularity of the Lyapunov exponent for Schrödinger cocycles

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*schedule*
le mardi 18 juin 2019 de 10h30 à 12h00

**Organisé par :**David Burguet et Pierre-Antoine Guiheneuf

**Intervenant :**Thomas Ohlson Timoudas (KTH (Stockholm))

**Lieu :**salle 16.26.209

**Sujet :**Loss of uniform hyperbolicity and regularity of the Lyapunov exponent for Schrödinger cocycles

**Résumé :**

What happens when a system bifurcates from uniform, to
non-uniform, hyperbolic behaviour? This type of bifurcation happens for
Schrödinger cocycles with large forcing, and is connected to the
spectrum of the corresponding operator.

We start off by considering the cosine like class of quasi-periodically forced Schrödinger cocycles, with large forcing, for which there is very precise information about the stable and unstable directions. We will show how this information can be used to obtain regularity results for the Lyapunov exponent, as a function of the energy parameter.

In general, the Lyapunov exponent is expected to be essentially 1/2-Hölder continuous. Indeed, for certain analytic classes of Schrödinger operators, this is already known. However, in the smooth case, the results are few and not very sharp.

In a joint work with Jordi-Lluis Figueras, we have obtained square root like asymptotics for the Lyapunov exponent at the lowest energy of the spectrum for a general class of Schrödinger cocycles.

Our approach is divided into two parts. The first part applies to more general quasi-periodic cocycles, using an identity by Avila, and assumes certain estimates. The second part is specific to the Schrödinger class we consider, where the estimates are based on an inductive construction by Bjerklöv.

We start off by considering the cosine like class of quasi-periodically forced Schrödinger cocycles, with large forcing, for which there is very precise information about the stable and unstable directions. We will show how this information can be used to obtain regularity results for the Lyapunov exponent, as a function of the energy parameter.

In general, the Lyapunov exponent is expected to be essentially 1/2-Hölder continuous. Indeed, for certain analytic classes of Schrödinger operators, this is already known. However, in the smooth case, the results are few and not very sharp.

In a joint work with Jordi-Lluis Figueras, we have obtained square root like asymptotics for the Lyapunov exponent at the lowest energy of the spectrum for a general class of Schrödinger cocycles.

Our approach is divided into two parts. The first part applies to more general quasi-periodic cocycles, using an identity by Avila, and assumes certain estimates. The second part is specific to the Schrödinger class we consider, where the estimates are based on an inductive construction by Bjerklöv.

We will further argue that these methods can be extended to cover other parameter values, and other types of quasi-periodic cocycles.