A Sampling Theorem for Rotation Numbers of Linear Processes in ${\mathbb{R}}^2$

Abstract

We prove an ergodic theorem for the rotation number of the composition of a sequence os stationary random homeomorphisms in $S^1$. In particular, the concept of rotation number of a matrix $g\in G{l}^{+}(2,\mathbb{R})$ can be generalized to a product of a sequence of stationary random matrices in $G{l}^{+}(2,\mathbb{R})$. In this particular case this result provides a counter-part of the Osseledec's multiplicative ergodic theorem which guarantees the existence of Lyapunov exponents. A random sampling theorem is then proved to show that the concept we propose is consistent by discretization in time with the rotation number of continuous linear processes on ${\mathbb{R}}^2.$