Abstract:

In this talk I will discuss an abstract, dynamical Thouless-type formula for affine families of GL(2, R) cocycles. This result extends the classical formula relating, via the Hilbert transform, the maximal Lyapunov exponent and the integrated density of states of a Schrödinger operator. Here, the role of the integrated density of states will be played by a more geometrical quantity, the fibered rotation number. As an application of this formula we present limitations on the modulus of continuity of random linear cocycles. Moreover, we derive Hölder-type continuity properties of the fibered rotation number for linear cocycles over various base dynamics.

Based on joint work with J. Bezerra, A. Cai, P. Duarte and C. Freijo.