Weak mixing for logarithmic flows over interval exchange transformations
Corinna Ulcigrai
Journal of Modern Dynamics 2009, 3(1): 35-49 doi: 10.3934/jmd.2009.3.35
We consider a class of special flows over interval exchange transformations which includes roof functions with symmetric logarithmic singularities. We prove that such flows are typically weakly mixing. As a corollary, minimal flows given by multivalued Hamiltonians on higher-genus surfaces are typically weakly mixing.
keywords: logarithmic singularities area-preserving flows on surfaces. Weak mixing interval exchange transformations
Time-changes of horocycle flows
Giovanni Forni Corinna Ulcigrai
Journal of Modern Dynamics 2012, 6(2): 251-273 doi: 10.3934/jmd.2012.6.251
We consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate of mixing. We then derive results on the spectrum of smooth time-changes and show that the spectrum is absolutely continuous with respect to the Lebesgue measure on the real line and that the maximal spectral type is equivalent to Lebesgue.
keywords: Time-changes horocycle flows quantitative mixing spectral theory. quantitative equidistribution
Genericity on curves and applications: pseudo-integrable billiards, Eaton lenses and gap distributions
Krzysztof Frączek Ronggang Shi Corinna Ulcigrai
Journal of Modern Dynamics 2018, 12(1): 55-122 doi: 10.3934/jmd.2018004

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. In the space of affine lattices $ASL_2( \mathbb{R})/ASL_2( \mathbb{Z})$, we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum $\mathscr{H}(1,1)$ of translation surfaces. For these curves we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers is also obtained.

keywords: Homogeneous dynamics translation surfaces Kontsevich-Zorich cocycle equidistribution ergodic theorem Oseledets theorem Eaton lenses gap distribution pseudo-integrable billiards

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