ABSTRACT:
In 2001, M. Goldstein and W. Schlag introduced the Avalanche Principle, a quantitative sufficient condition for the operator norm $\|A_N\cdots A_1\|$ of a product of matrices in $\mathrm{SL}_2(\mathbb{R})$, to being similar to the product $\|AN\|\cdots \|A_1\|$. Since then several refinements and generalizations have appeared in the literature. In this talk I will present a reformulation of this principle in terms of the geometry of the hyperbolic plane, and show how to extend it to metric spaces of negative curvature.