Hyperbolic dynamical systems, invariant geometric structures, and rigidity

Renato Feres
1994 Mathematical Research Letters  
The problems and results described in this article are situated at a point of contact between differential geometry on the one hand and the theory of hyperbolic dynamical systems on the other. They are part of a varied and growing body of research that has generally been described as Geometric Rigidity. ([22] provides a fairly representative cross-section of the field.) We consider the following, for the moment vague, question: What can be said about the analytic and geometric properties (and
more » ... e existence) of geometric structures preserved under group actions that exhibit some form of hyperbolic behavior? By a differential geometric structure, I wish to understand that kind considered by M. Gromov in [6] , which he calls rigid geometric structure. A basic example of such structures, and the one considered here for the most part, is an affine connection. These rigid structures have in common that their pseudo-group of local automorphisms is finite dimensional. (That is to be contrasted, say, with a symplectic structure. This notion generalizes Cartan's structures of finite type.) The present discussion can be regarded as a specialization of some of the general questions formulated in [6] . The theory of smooth dynamical systems with hyperbolic behavior has developed, since the seminal work of D. V. Anosov [1], into a vast and rich body of pure as well as applied mathematics. I refer to [15] and [20] for general information on this subject. More recently, the theory has been applied with great profit to the study of actions of higher rank groups (see [18], [8]). The concept of rigidity is understood here as the property that certain classes of group actions possess that permit them to be classified, under some suitable notion of equivalence, by a small number of algebraically defined model actions. Usually, these model actions are given as automorphisms of double coset spaces of the form Γ\G/H, where H is a closed subgroup of the Lie group G and Γ is a discrete subgroup. Important examples of rigidity theories that are related to our discussion are the above
doi:10.4310/mrl.1994.v1.n1.a2 fatcat:xtcjxeq7rffq7ihww2ed734evi