Lyapunov exponents, the problem of regularity

The study of Lyapunov exponent goes back to the stability theory for differential equations developed in the doctoral thesis of Aleksandr M. Lyapunov, in the late 19th century, since then, it grew into a very broad area and active field in ergodic theory and dynamical systems, with several outstanding problems and applications.  In the early 80’s, Ricardo Mañé observed that the Lyapunov exponents of continuous 2-dimensional cocycle can be cancelled by arbitrarily small perturbation of the cocycle. The proof of this observation was completed by Jairo Bochi.

In 2010, Carlos Bocker and Marcelo Viana have proven that the Lyapunov exponent of random 2-dimensional cocycle vary always continuously with respect to the probability distribution. This result was also extended in arbitrarily dimensional by Artur Avila, Alex Eskin and Marcelo Viana. Recently Marcelo Viana and myself have proven that the Lyapunov exponents of random 2-dimensional cocycle are Hölder continuous functions of the underlying probability distribution at each point with simple Lyapunov spectrum. Moreover, they are log-Hölder continuous at every point.

I will first discuss the problem of continuity and give some ideas and results about Hölder continuity.