## Lebesgue spectrum of countable multiplicity for conservative flows on the torus

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Bassam Fayad, Giovanni Forni and Adam Kanigowski
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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.
- © Copyright 2021 American Mathematical Society
- 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
- MathSciNet review: 4334191