Density of positive Lyapunov exponents for 𝑆𝐿(2,ℝ)-cocycles

Abstract

We show that S L ( 2 , R ) \mathrm {SL}(2,\mathbb {R}) -cocycles with a positive Lyapunov exponent are dense in all regularity classes and for all non-periodic dynamical systems. For Schrödinger cocycles, we show prevalence of potentials for which the Lyapunov exponent is positive for a dense set of energies.