Probabilistic averages of Jacobi operators

Abstract

I study the Lyapunov exponent and the integrated density of states for general Jacobi operators. The main result is that questions about these, can be reduced to questions about ergodic Jacobi operators. Then, I apply this to $a(n)=1$ and $b(n)=f(n^{\rho }(\mathrm{mod}1))$ for $\rho >0$ not an integer, and to obtain a probabilistic version of the Denisov--Rakhmanov--Remling Theorem.

References in corpus (3)

• Schrödinger Operators Defined by Interval Exchange Transformations
• A family of Schrödinger operators whose spectrum is an interval
• Reflectionless Herglotz functions and generalized Lyapunov exponents

Cited by in corpus (6)

• An explicit skew-shift Schrödinger operator with positive Lyapunov exponent at small coupling
• Finite Gap Jacobi Matrices: A Review
• Discontinuity of the Lyapunov Exponent
• Kotani-Last problem and Hardy spaces on surfaces of Widom type
• Spectral Aspects of the Skew-Shift Operator: A Numerical Perspective
• Orthogonal polynomials on the unit circle with Verblunsky coefficients defined by the skew-shift