Angular values of nonautonomous and random linear dynamical systems: Part I  Fundamentals
Abstract
We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on longterm averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For onedimensional subspaces in twodimensional systems our angular values agree with the classical theory of rotation numbers for orientationpreserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than $\frac{\pi}{2}$. Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.11305
 Bibcode:
 2020arXiv201211305B
 Keywords:

 Mathematics  Dynamical Systems;
 37C05;
 37E45;
 37A05;
 65Q10;
 15A18