The limit set of a discrete group of hyperbolic motions

Abstract

Consider a discrete group Γ of Mobius transformations acting in the unit ball of R n (n ≥ 2). The limit set of T is that subset of the unit sphere where T orbits accumulate and, as such, is the set of points where T fails to act discontinuously. Over the last several years much work has been done on the classification of limit points—a major impetus in this direction has been provided by the application of ergodic theory to discrete groups. Put simply, in order to understand the dynamics of the flow along a geodesic, for example, one must have information about the (limit) point situated at the “end” of the geodesic. In order to obtain general theorems about the classes of groups for which certain flows exhibit ergodic properties, one needs information on the size of various subsets of the limit set. Similarly the existence, or otherwise, of wandering sets under the group action on the sphere depends solely on the size of some other subsets of the limit set. Many subclasses of the limit set have been identified and their properties explored. The literature is now very extensive and, due to the differing definitions and notations, is somewhat confusing.KeywordsLimit PointHausdorff DimensionDiscrete GroupKleinian GroupFuchsian GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.