Reflectionless Herglotz functions and generalized Lyapunov exponents

TL;DR

This paper explores reflectionless Jacobi matrices, focusing on their measures, Lyapunov exponents, and density of states, providing new insights into their spectral properties and related quantities.

Contribution

It introduces a general framework for Lyapunov exponents and density of states measures in the context of reflectionless Jacobi matrices, extending previous understanding.

Findings

Analysis of the singular part of reflectionless measures

Introduction of generalized Lyapunov exponents

Connection between density of states measures and reflectionless properties

Abstract

We study several related aspects of reflectionless Jacobi matrices. Our first set of results deals with the singular part of reflectionless measures. We then introduce and discuss Lyapunov exponents, density of states measures, and other related quantities in a \textit{general} setting. This is related to the previous material because the density of states measures are reflectionless on certain sets.