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Lebesgue spectrum of countable multiplicity for conservative flows on the torus


Authors: Bassam Fayad, Giovanni Forni and Adam Kanigowski
Journal: J. Amer. Math. Soc. 34 (2021), 747-813
MSC (2020): Primary 37A25, 37A30, 37E35; Secondary 37C10, 37D40
DOI: https://doi.org/10.1090/jams/970
Published electronically: March 25, 2021
MathSciNet review: 4334191
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Abstract:

We study the spectral measures of conservative mixing flows on the $2$-torus having one degenerate singularity. We show that, for a sufficiently strong singularity, the spectrum of these flows is typically Lebesgue with infinite multiplicity.

For this, we use two main ingredients: (1) a proof of absolute continuity of the maximal spectral type for this class of non-uniformly stretching flows that have an irregular decay of correlations, (2) a geometric criterion that yields infinite Lebesgue multiplicity of the spectrum and that is well adapted to rapidly mixing flows.


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Additional Information

Bassam Fayad
Affiliation: CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, France
MR Author ID: 675142

Giovanni Forni
Affiliation: Department of Mathematics, University of Maryland, College Park, College Park, MD 20742
MR Author ID: 308447

Adam Kanigowski
Affiliation: Department of Mathematics, University of Maryland, College Park, College Park, MD 20742
MR Author ID: 995679

Received by editor(s): May 29, 2019
Received by editor(s) in revised form: April 7, 2020, August 19, 2020, and November 2, 2020
Published electronically: March 25, 2021
Additional Notes: The first author was supported by ANR-15-CE40-0001 and by the project BRNUH. The second author was supported by NSF Grants DMS 1201534 and 1600687, and by a Simons Fellowship.
Article copyright: © Copyright 2021 American Mathematical Society