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$ 1\ 2\ \ldots \ m-1 \ m $

$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ m-1 \ldots \ 2\ 1 $
for m=3,5,7.

Some disjointness properties of special flows built over interval exchange transformations and under piecewise constant roof function are investigated as well.

$\mathbb T^3\quad (x_1,x_2,x_3)\mapsto (x_1+\alpha,\varepsilon x_2+\beta(x_1),x_3+\gamma(x_1,x_2))\in\mathbb T^3,

where $\varepsilon =\pm 1$. We also indicate why there is no $4$-dimensional analogue of the above result. Random diffeomorphisms on the $2$-torus are studied as well.

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.

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