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

TL;DR

This paper establishes an ergodic theorem for rotation numbers of sequences of stationary random matrices in two dimensions, generalizing classical concepts and providing a sampling theorem linking discrete and continuous linear processes.

Contribution

It introduces a new ergodic theorem for rotation numbers of stationary random matrices, extending classical rotation number concepts and connecting discrete and continuous processes.

Findings

Proves an ergodic theorem for rotation numbers of stationary random homeomorphisms.

Generalizes rotation number concept to products of stationary random matrices.

Establishes a sampling theorem linking discrete rotation numbers to continuous linear processes.

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 $g\in Gl^{+}(2,{\R})$ can be generalized to a product of a sequence of stationary random matrices in $Gl^{+}(2,{\R})$. 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 ${\R}^{2}.$