Johnson-Schwartzman Gap Labelling for Ergodic Jacobi Matrices

David Damanik; Jake Fillman; Zhenghe Zhang

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arXiv · 2208.01143

Johnson-Schwartzman Gap Labelling for Ergodic Jacobi Matrices

David Damanik·Jake Fillman·Zhenghe Zhang

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Abstract

We consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphim of a compact metric space. Given an ergodic probability measure, we study the topological structure of the associated almost sure spectrum. We establish a gap labelling theorem in the spirit of Johnson and Schwartzman. That is, we show that the constant value the integrated density of states takes in a gap of the spectrum must belong to the countable Schwartzman group of the base dynamics. This result is a natural companion to a recent result of Alkorn and Zhang, which established a Johnson-type theorem for the families of Jacobi matrices in question.

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BibTeX RIS

David Damanik, Jake Fillman, Zhenghe Zhang. 2022-08-01. Johnson-Schwartzman Gap Labelling for Ergodic Jacobi Matrices. https://arxiv.org/abs/2208.01143

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