Estimates on Grassmann Manifolds
Pedro Duarte, Silvius Klein · Atlantis Press eBooks · 2016
The main result of this chapter, 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 \(\mathrm{SL}(2,\mathbb {C})\) matrices. Besides extending the AP to matrices of arbitrary dimension and 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 the next chapters, when we establish the continuity the LE and of the Oseledets decomposition. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.