scholarly journals Ratner's Work on Unipotent Flows and Impact

2019 ◽  
Vol 66 (03) ◽  
pp. 1
Author(s):  
Elon Lindenstrauss ◽  
Peter Sarnak ◽  
Amie Wilkinson
Keyword(s):  
Unipotent Flows

scholarly journals On orbits of unipotent flows on homogeneous spaces, II

1986 ◽  
Vol 6 (2) ◽  
pp. 167-182 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Invariant Subspace ◽  
Lebesgue Measure ◽  
Compact Subset ◽  
Lie Group ◽  
Diophantine Approximation ◽  
Closed Subgroup ◽  
Homogeneous Spaces ◽  
Semisimple Lie Group ◽  
Unipotent Subgroup ◽  
Unipotent Flows

AbstractWe show that if (ut) is a one-parameter subgroup of SL (n, ℝ) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, ℝ)/SL(n, ℤ) such that the following holds: for any g ∈ SL(n, ℝ) either m({t ∈ [0, T] | utg SL (n, ℤ) ∈ K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations.Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation.

scholarly journals Corrections to the paper ‘On orbits of unipotent flows on homogeneous spaces’

1986 ◽  
Vol 6 (2) ◽  
pp. 321-321
Author(s):  
S. G. Dani
Keyword(s):  
Homogeneous Spaces ◽  
Unipotent Flows

The author regrets that there are certain errors in [1] and would like to give the following corrections.

scholarly journals Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

2017 ◽  
Vol 38 (7) ◽  
pp. 2780-2800 ◽  
Author(s):  
RODOLPHE RICHARD ◽  
NIMISH A. SHAH
Keyword(s):  
Number Theory ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Reductive Groups ◽  
Natural Group ◽  
Homogeneous Dynamics ◽  
Geometric Conditions ◽  
Finite Dimensional ◽  
Unipotent Flows ◽  
Linearization Techniques

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 vector spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.

On unipotent flows in ℋ(1,1)

2009 ◽  
Vol 30 (2) ◽  
pp. 379-398 ◽  
Author(s):  
KARIANE CALTA ◽  
KEVIN WORTMAN
Keyword(s):  
Moduli Space ◽  
Probability Measures ◽  
Specific Class ◽  
Horocycle Flow ◽  
Abelian Differentials ◽  
Unipotent Flows ◽  
Genus Two

AbstractWe study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures that are invariant and ergodic under the horocycle flow on the stratum ℋ(1,1).

Minimal sets of homogeneous flows

1995 ◽  
Vol 15 (2) ◽  
pp. 361-377 ◽  
Author(s):  
A. N. Starkov
Keyword(s):  
Homogeneous Space ◽  
Finite Volume ◽  
Orbit Closure ◽  
Sufficient Condition ◽  
Minimal Sets ◽  
Unipotent Flows ◽  
Topological Theorem

AbstractGeneralizing Ratner's topological theorem for unipotent flows we obtain a sufficient condition for an orbit closure of a flow on a homogeneous space of finite volume to be algebraic. We give some applications of the result to studying birecurrent and minimal sets of homogeneous flows.

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