The avalanche principle and negative curvature

Abstract

We use the geometric structure of the hyperbolic upper half plane to provide a new proof of the Avalanche Principle introduced by M. Goldstein and W. Schlag in the context of $$\mathrm {SL}_{2}({\mathbb {R}})$$ matrices. This approach allows to interpret and extend this result to arbitrary $$\text {CAT}(-1)$$ metric spaces. Through the proof, we deduce a polygonal Schur theorem for these spaces.