Dynamical Systems with Transitive Symmetry Group. Geometric and Statistical Properties

  • A. V. Safonov
  • A. N. Starkov
  • A. M. Stepin
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 66)


The main topic of discussion in this survey is the geometric and metric properties of geodesic and horocycle flows on surfaces of constant curvature and their multidimensional analogues, the so-called G-induced actions (that is, restrictions to a suitable subgroup F of the natural action of G on a homogeneous space of it). One usually takes as F one-parameter or cyclic subgroups of G and the horospherical subgroups corresponding to them (the details are given in Sect. 1). However, in the course of the exposition of our main topic we sometimes mention results relating to the case when F is an n-parameter subgroup of G or a lattice. This class of dynamical systems consists precisely of group actions admitting an extension to a transitive action of a Lie group. We shall not be considering in our survey (or, at any rate, not systematically) the following themes which are close to our main subject matter:
  • Topological dynamics and metric properties of general group actions (in particular, the work of Zimmer on semisimple group actions (Zimmer 1984);

  • Group automorphisms;

  • Affine transformations of homogeneous spaces;

  • G-induced flows on spaces with an infinite invariant measure. The papers (Hopf 1939), (Sullivan 1981) and the recent book (Nicholls 1989) are devoted to the connections between G-induced flows (and also geodesic flows) and the theory of discrete groups of motions of a hyperbolic space;

  • The actions of subgroups on the double cosets related to geodesic (for n > 2) and horocycle (for n > 3) flows on an n-dimensional manifold of constant negative curvature (see (Gel’fand and Fomin 1952) and, in particular, (Mautner 1957)). Flows of this type in somewhat more general form differing from geodesic and horocycle flows have not been investigated.


Homogeneous Space Geodesic Flow Homogeneous Flow Constant Negative Curvature Unique Ergodicity 
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© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • A. V. Safonov
  • A. N. Starkov
  • A. M. Stepin

There are no affiliations available

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