Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces

By M. Bekka, M. Mayer. 2000

217 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9e62c44d5baa7a1717813704756d5b0b09c09f3e

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Ratner's Theorems on Unipotent Flows (2003, unknown): “(See [1] for an elementary explanation.”
On effective equidistribution of expanding translates of certain orbits in the space of lattices (2007, unknown): “∀v∈ Zkr{0} . In other words, Kε consists of lattices in Rk with no nonzero vector of length less than ε. These sets are compact by virtue of Mahler’s Compactness Criterion (see [R, Corollary 10.9] or [BM]). Here k·k can be any norm on Rkwhich we will from now on take to be the standard Euclidean nor”
Badly approximable systems of affine forms, fractals, and Schmidt games (2009, methodology): “The proofs of our results require two tools: Schmidt games (see [17] for a reference) and the basic concepts concerning flows on the space of unimodular lattices (see Chapter 9 of [6] or [1] for a reference).”
Chapter 11 Dynamics of subgroup actions on homogeneous spaces of lie groups and applications to number theory (2002, background): “An elementary proof (not involving representation theory) of the theorem was sketched by Margulis in [142]; for details see [13,237].”
Continued fractions for complex numbers and values of binary quadratic forms (2011, background): “This implies in particular that the action of D on SL(2,C)/SL(2,G) is ergodic (with respect to the SL(2,C)-invariant measure); see [2], Chapter III, for instance.”
A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces (2019, background): “for example, [BM00, Lemma 3.6]), is that the action of u t= 1 t 0 1 on Q 1 is also ergodic. Another corollary, which follows from a general result called the Howe-Moore Theorem (see, for example, [BM00]), is that the action of g tis not just ergodic but mixing. This means that, not only do orbit s”
Isometric Group Actions on Banach Spaces and Representations Vanishing at Infinity (2006, background): “π. Since the G-action on X is mixing, the representation π is C0 (see [ BM ]), so that the conclusion follows immediately from Property (BP0).”