Abstract Continuity Theorem for the Lyapunov Exponents of linear cocycles

Abstract: We prove the H\"older continuity of Lyapunov exponents for general linear cocycles when the base measures vary in Wasserstein distance, under the assumption of uniform large deviations type (LDT) estimates. This is a measure version of the abstract continuity theorem (ACT) established by Duarte-Klein [Duarte, P. and Klein, S. (2016). Lyapunov exponents of linear cocycles: Continuity via large deviations. Atlantis Studies in Dynamical Systems, 3]. The main obstacle here lies in the fact that the magnitude of the exceptional sets in LDT estimates is constantly changing when the base measures deviate. We overcome this via a combination of a Urysohn-type lemma and properties of Wasserstein distance in every iteration step. Our measure version of ACT, combined with the original work of Duarte-Klein, provides a complete scheme for proving joint H\"older continuity of Lyapunov exponents with respect to both measure and fiber which resolves all parameter dependence. This continuity theorem is general and applicable to a wide range of mathematical models, including product of random matrices and cocycles essentially generated by shifts. In particular, it applies to associated Schr\"odinger operators which are central objects in the study of mathematical physics.