On the Correspondence Between Domination and the Spectrum of Jacobi Operators

TL;DR

This paper establishes a connection between the spectrum of Jacobi operators and the absence of dominated splitting in associated cocycles, generalizing previous results for Schrödinger and dynamically defined operators.

Contribution

It introduces a notion of dominated splitting for matrix sequences and proves its stability, linking spectral properties of Jacobi operators to cocycle dynamics, extending classical theorems.

Findings

Spectrum corresponds to non-dominated splitting of Jacobi cocycles.

Generalizes Johnson's theorem for Schrödinger operators.

Includes dynamically defined Jacobi operators with transitive base dynamics.

Abstract

In this paper, we first develop a notion of dominated splitting for $\mathbb M(2,\mathbb C)$-sequences and show it is a stable property under $\|\cdot \|_\infty$-perturbation. Then we show an energy parameter belongs to the spectrum of a Jacobi operator, possibly singular, if and only if the associated Jacobi cocycle does not admit dominated splitting. This generalizes the results obtained by the second author [Z] in the scenario of Schr\"odinger operators. Finally, we consider dynamically defined Jacobi operators whose base dynamics is only assumed to be topologically transitive. We show an energy parameter belongs to the spectrum of the operator defined by the base point with a dense orbit if and only if the dynamically defined Jacobi cocycle does not admit dominated splitting. This includes the original Johnson's theorem obtained by R. Johnson [J] for Sch\"odinger operators and the main theorem obtained by C. Marx [Ma] for Jacobi operators as special cases.