On SL(2,R)-Cocycles over Irrational Rotations with Secondary Collisions

Article Mathematics

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

Anton Gorodetski et al.

Summary: The study involves random products of SL(2,R) matrices dependent on a parameter in a non-uniformly hyperbolic regime. It shows that under monotone parameter dependence, the Lyapunov exponents of the random product satisfy specific conditions, and provides a purely geometrical proof of Spectral Anderson Localization for discrete Schrodinger operators on a one-dimensional lattice with random potentials.

ADVANCES IN MATHEMATICS (2021)

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Article Mathematics, Applied

On Singularly Perturbed Linear Cocycles over Irrational Rotations

Alexey V. Ivanov

Summary: The study focuses on the properties of a linear cocycle on the circle T-1 generated by a C-2 map A(epsilon) depending on a small parameter epsilon and having the form of a Poincare map. It examines the behavior of the cocycle with an assumption on the norm of the matrix A(epsilon)(x) and discusses its exponential dichotomy with respect to the parameter epsilon. The results suggest that the cocycle typically exhibits an exponential dichotomy only when it is exponentially close to a constant cocycle in the limit epsilon -> 0.

REGULAR & CHAOTIC DYNAMICS (2021)

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Article Mathematics, Applied