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# Séminaire DynamiX

École polytechnique – Centre de Physique Théorique, Salle de conférences Jean Lascoux (Aile 0)

14h00-16h00 – Giovanni Forni (Univ. of Maryland)

"Countable Lebesgue spectrum for some smooth parabolic flows"

The spectrum of a dynamical systems encodes information on the Fourier transforms of correlations of  square-integrable observables. In classical ergodic theory it was proved that if a dynamical system has the K property, then it has countable Lebesgue spectrum

We introduce a criterion for countable Lebesgue spectrum which can be applied to smooth (parabolic) flows with zero entropy, and prove that

1) the generic smooth (Kochergin) flow on the $2$-torus with a single sufficiently degenerate rest point  and

2) smooth time-changes of horocycle flows.have countable Lebesgue spectrum (as conjectured by Katok-Thouvenot).

For the latter case, we had proven a few years ago with Ulcigrai that the maximal spectral type is absolutely continuous.

The new results are joint work with Bassam Fayad and Adam Kanigowski.