BOUNDED ORBITS OF NONQUASIUNIPOTENT FLOWS ON HOMOGENEOUS SPACES

By Dmitry Kleinbock, Dani Dan. 2002

235 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a6266e23fdf13108b7a4825932366ff822687225

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Invariant distributions and time averages for horocycle flows (2003, background): “Kleinbock [31] proved, for finite volume quotients of general semisimple Lie groups, exponential decay of the deviation from equidistribution for smooth measures supported on a compact subset of any horosphere under the action of diagonal subgroups.”
Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices (2005, methodology): “This is always possible (compare with Lemma 2.4.7 of [ KM ]).”
Matrix coefficients, counting and primes for orbits of geometrically finite groups (2012, background): “Let H†δ denote the sum of of all complementary series representations of parameter δ contained in L2(0\G), and let Pδ denote the projection operator from L2(0\G) to H†δ .”
Two-dimensional badly approximable vectors and Schmidt's game (2012, background): “Kleinbock observed that if F 1 and F + 2 lie in opposite Weyl chambers, the arguments in [10] can be adapted to prove that the set of g ∈ G satisfying the conjecture has full Hausdorff dimension.”
The distribution of periodic torus orbits on homogeneous spaces (2006, background): “The following is a simple special case of a very general fact, see [23] and the references therein.”
Hausdorff dimension of the set of singular pairs (2007, background): “The set of vectors in Q that realise some local minimum of Wx will be denoted by Σ(x).”
Equidistribution and counting for orbits of geometrically finite hyperbolic groups (2010, background): “9 is due to Sarnak [39] for horocycles in H2 and Duke-Rudnick-Sarnak [9] and Eskin-McMullen [11] in general (see also the appendix of [17]).”
Limit theorems for partially hyperbolic systems (2003, background): “[20, 53] produce many non-negative C∞ functions for which An ≡ 0 on a set of large Hausdorff dimension.”
Badly approximable vectors on fractals (2005, methodology): “As a historical remark, let us point out that Dani in [D1] used the correspondence of Proposition 3.1 and Schmidt’s result on the full Hausdor dimension of the set BA to derive (3.6); and that later a dynamical proof of a generalization of (3.6) appeared in [ KM1 ].”
Averaging and Invariant Measures (2005, unknown): “(1) Anosov diffeomorphisms [7]; (2) time one maps of Anosov flows [15], [34], [22]; (3) partially hyperbolic translations on homogeneous spaces [32]; (4) compact group extensions of Anosov diffeomorphisms [17]; (5) partially hyperbolic toral automorphisms [28]; (6) mostly contracting systems (see Se”
Modified Schmidt games and Diophantine approximation with weights (2008, methodology): “7 we will use a construction suggested in [17,24] and formalized in [10].”
DIRICHLET'S THEOREM ON DIOPHANTINE APPROXIMATION AND HOMOGENEOUS FLOWS (2006, background): “These conditions were shown in [15] to imply that for any x ∈G/Γ, the set { h ∈ H : the trajectory {gt hx : t > 0} is bounded }”