| Abstract: |
| We study the quantitative regularity of Lyapunov exponents for non-compact
and non-invertible random linear cocycles, with respect to the Wasserstein distance. Our first
main result shows that, under three natural hypotheses (finite exponential moments,
quasi-irreducibility, and a spectral gap $L_1 > L_2$) the top Lyapunov exponent depends
locally H\{o}lder-continuously on the measure that governs the dynamics. The proof relies on
a spectral method in which the strong mixing of the associated Markov operator is crucial.
To that end, we extend Furstenberg-Kifer theory beyond the invertible setup. Consequences
include H\{o}lder continuity of higher exponents via exterior powers, large-deviations--type
estimates, and applications to Schr\{o}dinger cocycles with unbounded potentials.
Our second main result shows that the exponential moment assumption is sharp. We
construct non-compact random Schr\{o}dinger cocycles with locally uniform sub-exponential
moments but without any exponential moment, for which the Lyapunov exponent, as a function
of the energy, fails to be $\alpha$-H\{o}lder for every $\alpha > 0$. More generally, we
provide a correspondence between moment profiles and moduli of continuity, and we prove that
the breakdown of a given (locally uniform) moment condition prevents the corresponding
modulus of continuity from holding. |
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