CAPA-seminarium: Loss of uniform hyperbolicity and regularity of the Lyapunov exponent

  • Datum:
  • Plats: Ångströmlaboratoriet 64119
  • Föreläsare: Thomas Olson
  • Kontaktperson: Alejandro Luque
  • Seminarium

Abstract: What happens when a system undergoes a bifurcation from uniform, to non-uniform, hyperbolic behaviour? We consider a certain class of smooth, Diophantine, quasi-periodically forced Schrödinger cocycles, where we can study how the stable and unstable directions break apart under the dynamics. This information can then be used to obtain regularity results for the Lyapunov exponent.

The parameter E we consider is related to the spectrum of the corresponding Schrödinger operator. Almost optimal regularity results, for the Lyapunov exponent L(E) depending on this parameter, are already known for certain classes of analytic Schrödinger cocycles, but in the smooth case the results are fewer and typically not as sharp. For certain parameter values (the lowest energy of the spectrum) we obtain good estimates on the regularity of the Lyapunov exponent, as long the cocycle satisfies a non-degeneracy condition (which is usually imposed to guarantee hyperbolicity).

We will argue that these methods can be extended to cover other parameter values, and even other types of dynamical systems.