The Avalanche Principle and other estimates on Grassmann manifolds

TL;DR

This paper introduces the Avalanche Principle (AP) for matrices of any dimension, relating the expansion of long matrix products to individual matrices, with applications to Lyapunov exponents and Oseledets decomposition.

Contribution

It extends the Avalanche Principle to arbitrary dimension matrices, including non-invertible ones, and connects singular directions of matrix products with those of individual matrices.

Findings

Extended AP to non-invertible matrices.

Established relations between singular directions of products and factors.

Provided estimates on matrix actions on Grassmann manifolds.

Abstract

The main result of this paper, called the Avalanche Principle (AP), relates the expansion of a long product of matrices with the product of expansions of the individual matrices. This principle was introduced by M. Goldstein and W. Schlag in the context of ${\rm SL}(2,\mathbb{C})$ matrices. Besides extending the AP to matrices of arbitrary dimension, possibly non-invertible, the geometric approach we use here provides a relation between the most expanding (singular) directions of such a long product of matrices and the corresponding singular directions of the first and last matrices in the product. The AP along with other estimates on the action of matrices on Grassmann manifolds will play a fundamental role in [6] to establish the continuity the Lyapunov exponents and of the Oseledets decomposition for linear cocycles. This is the draft of a chapter in our forthcoming research monograph [6].