Lebesgue spectrum of countable multiplicity for conservative flows on the torus

Fayad, Bassam; Forni, Giovanni; Kanigowski, Adam

doi:10.1090/jams/970

Lebesgue spectrum of countable multiplicity for conservative flows on the torus
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by Bassam Fayad, Giovanni Forni and Adam Kanigowski

J. Amer. Math. Soc. 34 (2021), 747-813

DOI: https://doi.org/10.1090/jams/970

Published electronically: March 25, 2021

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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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