186 research outputs found
Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms
We show that a multiplicative form of Dirichlet's theorem on simultaneous
Diophantine approximation as formulated by Minkowski, cannot be improved for
almost all points on any analytic curve on R^k which is not contained in a
proper affine subspace. Such an investigation was initiated by Davenport and
Schmidt in the late sixties.
The Diophantine problem is then settled by showing that certain sequence of
expanding translates of curves on the homogeneous space of unimodular lattices
in R^{k+1} gets equidistributed in the limit. We use Ratner's theorem on
unipotent flows, linearization techniques, and a new observation about
intertwined linear dynamics of various SL(m,R)'s contained in SL(k+1,R).Comment: 28 page
Limiting distributions of curves under geodesic flow on hyperbolic manifolds
We consider the evolution of a compact segment of an analytic curve on the
unit tangent bundle of a finite volume hyperbolic -manifold under the
geodesic flow. Suppose that the curve is not contained in a stable leaf of the
flow. It is shown that under the geodesic flow, the normalized parameter
measure on the curve gets asymptotically equidistributed with respect to the
normalized natural Riemannian measure on the unit tangent bundle of a closed
totally geodesically immersed submanifold.
Moreover, if this immersed submanifold is a proper subset, then a lift of the
curve to the universal covering space is mapped into a proper
subsphere of the ideal boundary sphere under the visual map.
This proper subsphere can be realized as the ideal boundary of an isometrically
embedded hyperbolic subspace in covering the closed immersed submanifold.
In particular, if the visual map does not send a lift of the curve into a
proper subsphere of , then under the geodesic flow the curve gets
asymptotically equidistributed on the unit tangent bundle of the manifold with
respect to the normalized natural Riemannian measure.
The proof uses dynamical properties of unipotent flows on finite volume
homogeneous spaces of SO(n,1).Comment: 27 pages, revised version, Proof of Theorem~3.1 simplified, remarks
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Counting integral matrices with a given characteristic polynomial
We give a simpler proof of an earlier result giving an asymptotic estimate
for the number of integral matrices, in large balls, with a given monic
integral irreducible polynomial as their common characteristic polynomial. The
proof uses equidistributions of polynomial trajectories on SL(n,R)/SL(n,Z),
which is a generalization of Ratner's theorem on equidistributions of unipotent
trajectories. We also compute the exact constants appearing in the above
mentioned asymptotic estimate
Counting visible circles on the sphere and Kleinian groups
For a circle packing P on the sphere invariant under a geometrically finite
Kleinian group, we compute the asymptotic of the number of circles in P of
spherical curvature at most which are contained in any given region.Comment: Main results are significantly improved, 16 pages, 1 figur
Equidistribution and Counting for orbits of geometrically finite hyperbolic groups
Let G be the identity component of SO(n,1), acting linearly on a finite
dimensional real vector space V. Consider a vector w_0 in V such that the
stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line
Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup
Gamma of G with w_0Gamma discrete, we compute an asymptotic formula for the
number of points in w_0Gamma of norm at most T, provided that the
Bowen-Margulis-Sullivan measure on the associated hyperbolic manifold and the
Gamma skinning size of w_0 are finite.
The main ergodic ingredient in our approach is the description for the
limiting distribution of the orthogonal translates of a totally geodesically
immersed closed submanifold of Gamma\H^n. We also give a criterion on the
finiteness of the Gamma skinning size of w_0 for Gamma geometrically finite.Comment: Extensions of Equidistribution results to G/Gamma are obtained for
Gamma Zariski dense, and Much more precise description on the structure of
cuspidal neighborhoods of parabolic points is obtained for Gamma
geometrically finite. 63 pages (with 1 figure
Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics
Several problems in number theory when reformulated in terms of homogenous
dynamics involve study of limiting distributions of translates of algebraically
defined measures on orbits of reductive groups. The general non-divergence and
linearization techniques, in view of Ratner's measure classification for
unipotent flows, reduce such problems to dynamical questions about linear
actions of reductive groups on finite dimensional vectors spaces. This article
provides general results which resolve these linear dynamical questions in
terms of natural group theoretic or geometric conditions
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