Johnson–Schwartzman gap labelling for ergodic Jacobi matrices
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We consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphism on 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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Damanik, D., Fillman, J., & Zhang, Z. (2023). Johnson–Schwartzman gap labelling for ergodic Jacobi matrices. Journal of Spectral Theory, 13(1), 297–318. https://doi.org/10.4171/jst/449