On Point Spectrum of Jacobi Matrices Generated by Iterations of Quadratic Polynomials

Abstract

In general, point spectrum of an almost periodic Jacobi matrix can depend on the element of the hull. In this paper, we study the hull of the limit-periodic Jacobi matrix corresponding to the equilibrium measure of the Julia set of the polynomial \(z^2-\lambda \) with large enough \(\lambda \); this is the leading model in inverse spectral theory of ergodic operators with zero measure spectrum. We prove that every element of the hull has empty point spectrum. To prove this, we introduce a matrix version of Ruelle operators.