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

The Avalanche Principle and other estimates on Grassmann manifolds