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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Lebesgue spectrum of countable multiplicity for conservative flows on the torus
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by Bassam Fayad, Giovanni Forni and Adam Kanigowski HTML | PDF
J. Amer. Math. Soc. 34 (2021), 747-813 Request permission

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