Ergodic Jacobi matrices and conformal maps

TL;DR

This paper explores the relationship between ergodic Jacobi matrices and conformal maps, focusing on the Lyapunov exponent and density of states, extending classical analysis techniques to a broader framework.

Contribution

It introduces a general framework linking the Lyapunov exponent and density of states via conformal maps for ergodic Jacobi matrices, building on classical methods.

Findings

The function w=-γ + iπk acts as a conformal map between specific domains.

Structural properties of the Lyapunov exponent and density of states are characterized.

The approach generalizes classical analysis of periodic problems to ergodic settings.

Abstract

We study structural properties of the Lyapunov exponent $\gamma$ and the density of states $k$ for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function $w=-\gamma+i\pi k$ as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem.