Invariant Measures and Orbit Closures on Homogeneous Spaces for Actions of Subgroups Generated by Unipotent Elements

By Nimish A. Shah. 2000

64 citations recorded by the source corpus.

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and a closed subgroup of G. Let W be a subgroup of G such that AdG(W) is contained in the Zariski closure (in Aut(Lie G)) of the subgroup generated by the unipotent elements of AdG(W). Then any finite W-invariant W-ergodic measure on G/ is a homogeneous measure (i.e., it is supported on a closed orbit of a subgroup preserving the measure). Moreover, if G/ has finite volume (i.e., has a finite G-invariant measure), then the closure of any orbit of W on G/ is a homogeneous set (i.e., a finite volume closed orbit of a subgroup containing W). Both the above results hold if W is replaced by any subgroup � ⊂ W such that W/� has finite volume.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/02ea98efb901a0733d0c1d4682eeedc78df78dbf

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A non-conventional ergodic theorem for a nilsystem (2002, result): “Almost everywhere convergence for nilsystems was proved by Lesigne [11] and also follows from results of Shah [13].”
Stationary measures and invariant subsets of homogeneous spaces (III) (2013, background): “Corollary 1.6 is an analogue of the main theorem of Mozes and Shah in [20] (see also [12]) which asserts, in case G is a real Lie group, if E is the space of finite volume homogeneous subsets of X which are invariant and ergodic under some Ad-unipotent one-parameter subgroup of G, then the set E ∪ {”
The quantitative behaviour of polynomial orbits on nilmanifolds (2007, background): “As a corollary we obtain a Ratner-Shah type theorem for polynomial sequences in nilmanifolds, first established by Leibman [22]: Corollary 1.13 (Leibman’s Ratner-Shah type theorem for nilmanifolds).”
Some open problems on multiple ergodic averages (2011, unknown): “Equidistribution on nilmanifolds and other nil-stuff: [17, 57, 88, 89, 90, 110, 109, 122, 131, 169, 170, 171, 175, 176, 177, 187, 188, 190, 192, 195, 196, 210, 225, 227, 228, 230, 231, 232, 233, 234, 235, 241, 242, 246, 256, 257, 258, 259, 271, 272, 273, 285, 289, 299].”
Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold (2004, methodology): “These results are a very special case of general theorems of Ratner and Shah ([R], [Sh2]), but using a method of Parry ([P1] and [P2]) we can obtain a simple and independent proof thereof.”
Distribution of values of bounded generalized polynomials (2007, result): “It is also shown in [L3] (and, as well, follows from the results in [Le] or [Sh2]) that the closure Y = φ(Zd)x̃ of the orbit of”