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Uniformity aspects of ${\mathrm {SL}}(2,{\mathbb R})$ cocycles and applications to Schrödinger operators defined over Boshernitzan subshifts

Part of: Ergodic theory Special classes of linear operators

Published online by Cambridge University Press:  20 November 2024

DAVID DAMANIK Open the ORCID record for DAVID DAMANIK [Opens in a new window]  and
DANIEL LENZ
DAVID DAMANIK
Affiliation:
Department of Mathematics, Rice University, Houston, TX 77005, USA (e-mail: damanik@rice.edu)
DANIEL LENZ*
Affiliation:
Institute for Mathematics, Friedrich-Schiller University, Jena 07743, Germany
*
e-mail: daniel.lenz@uni-jena.de
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Abstract

We consider continuous ${\mathrm {SL}}(2,{\mathbb R})$ valued cocycles over general dynamical systems and discuss a variety of uniformity notions. In particular, we provide a description of uniform one-parameter families of continuous ${\mathrm {SL}}(2,{\mathbb R})$ cocycles as $G_\delta $-sets. These results are then applied to Schrödinger operators with dynamically defined potentials. In the case where the base dynamics is given by a subshift satisfying the Boshernitzan condition, we show that for a generic continuous sampling function, the associated Schrödinger cocycles are uniform for all energies and, in the aperiodic case, the spectrum is a Cantor set of zero Lebesgue measure.

Keywords

uniform cocyclesSchrödinger operatorssubshiftszero-measure spectrum

MSC classification

Primary: 37A20: Orbit equivalence, cocycles, ergodic equivalence relations 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 6 , June 2025 , pp. 1734 - 1756
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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Footnotes

Dedicated to the memory of Michael Boshernitzan

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