arXiv subjects
Publications and source records attributed to Tomé Graxinha.
In this paper, we present a class of random Schrödinger cocycles showing that, for random cocycles with non-compact support, the presence of certain finite moment conditions is essential for establishing a specific modulus of continuity of the Lyapunov exponent. In particular, Hölder continuity of the Lyapunov exponent requires an exponential moment condition.
Consider the space of two dimensional random linear cocycles over a shift in finitely many symbols, with at least one singular and one invertible matrix. We provide an explicit formula for the unique stationary measure associated to such cocycles and establish a Furstenberg-type formula characterizing the Lyapunov exponent. Using the spectral properties of the corresponding Markov operator and a parameter elimination argument, we prove that Lebesgue almost every cocycle in this space satisfies large deviations estimates and a central limit theorem.
We study the regularity of Lyapunov exponents for random linear cocycles taking values in $\Mat_m(\R)$ and driven by i.i.d. processes. Under three natural conditions - finite exponential moments, a spectral gap between the top two Lyapunov exponents, and quasi-irreducibility of the associated semigroup - we prove that the top Lyapunov exponent is Hölder continuous with respect to the Wasserstein distance. In the final section, we apply the main results to Schrödinger cocycles with unbounded potentials.
In this paper we establish a Bochi-Mañé type dichotomy in the space of two dimensional, nonnegative determinant matrix valued, locally constant linear cocycles over a Bernoulli or Markov shift. Moreover, we prove that Lebesgue almost every such cocycle has finite first Lyapunov exponent, which then implies a break in the regularity of the Lyapunov exponent, from analyticity to discontinuity.