Counting orbits of integral points in families of affine homogeneous varieties and diagonal flows
Alexander Gorodnik Frédéric Paulin
Journal of Modern Dynamics 2014, 8(1): 25-59 doi: 10.3934/jmd.2014.8.25
In this paper, we study the distribution of integral points on parametric families of affine homogeneous varieties. By the work of Borel and Harish-Chandra, the set of integral points on each such variety consists of finitely many orbits of arithmetic groups, and we establish an asymptotic formula (on average) for the number of the orbits indexed by their Siegel weights. In particular, we deduce asymptotic formulas for the number of inequivalent integral representations by decomposable forms and by norm forms in division algebras, and for the weighted number of equivalence classes of integral points on sections of quadrics. Our arguments use the exponential mixing property of diagonal flows on homogeneous spaces.
keywords: homogeneous variety counting norm form exponential decay of correlation. Siegel weight diagonalizable flow Integral point mixing decomposable form
Équidistribution, comptage et approximation par irrationnels quadratiques
Jouni Parkkonen Frédéric Paulin
Journal of Modern Dynamics 2012, 6(1): 1-40 doi: 10.3934/jmd.2012.6.1
Soit $M$ une variété hyperbolique de volume fini, nous montrons que les hypersurfaces équidistantes à une sous-variété $C$ de volume fini totalement géodésique s'équidistribuent dans $M$. Nous donnons une asymptotique précise du nombre de segments géodésiques de longueur au plus $t$, perpendiculaires communs à $C$ et au bord d'un voisinage cuspidal de $M$. Nous en déduisons des résultats sur le comptage d'irrationnels quadratiques sur $\mathbb{Q}$ ou sur une extension quadratique imaginaire de $\mathbb{Q}$, dans des orbites données des sous-groupes de congruence des groupes modulaires.

Let $M$ be a finite volume hyperbolic manifold. We show the equidistribution in $M$ of the equidistant hypersurfaces to a finite volume totally geodesic submanifold $C$. We prove a precise asymptotic formula on the number of geodesic arcs of lengths at most $t$, that are perpendicular to $C$ and to the boundary of a cuspidal neighbourhood of $M$. We deduce from it counting results of quadratic irrationals over $\mathbb{Q}$ or over imaginary quadratic extensions of $\mathbb{Q}$, in given orbits of congruence subgroups of the modular groups.
keywords: Equidistribution hyperbolic manifold quadratic irrational perpendicular geodesic. counting binary quadratic form
Escape of mass in homogeneous dynamics in positive characteristic
Alexander Kemarsky Frédéric Paulin Uri Shapira
Journal of Modern Dynamics 2017, 11(1): 369-407 doi: 10.3934/jmd.2017015

We show that in positive characteristic the homogeneous probability measure supported on a periodic orbit of the diagonal group in the space of $2$-lattices, when varied along rays of Hecke trees, may behave in sharp contrast to the zero characteristic analogue: For a large set of rays, the measures fail to converge to the uniform probability measure on the space of $2$-lattices. More precisely, we prove that when the ray is rational there is uniform escape of mass, that there are uncountably many rays giving rise to escape of mass, and that there are rays along which the measures accumulate on measures which are not absolutely continuous with respect to the uniform measure on the space of $2$-lattices.

keywords: Homogeneous measures positive characteristic lattices local fields escape of mass Hecke tree Bruhat-Tits tree equidistribution

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