Logarithm laws for one parameter 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.

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

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700003485 ◽

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.

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

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2016.138 ◽

2017 ◽

Vol 38 (7) ◽

pp. 2780-2800 ◽

Cited By ~ 1

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.

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