Parametric Furstenberg Theorem on random products of SL(2,R) matrices

By A. Gorodetski, V. Kleptsyn. Advances in Mathematics, 2021

50 citations recorded by the source corpus.

Abstract We consider random products of S L ( 2 , R ) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense G δ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrodinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4ee8c3e5e120e1a670834d1c48a8614776516fbd

Citations in the corpus

• Localization (2019, unknown): “Since then, various “single-scale” proofs of localization in one dimension which eschew MSA and leverage even further properties of random matrix products have appeared, see e.g. [JZ19, BDF + 17, GK21].”