Canonical connections for Anosov diffeomorphisms
by
Cem Tezer
Department of Mathematics, Middle East Technical University, 06531 Ankara-Turkey

An Anosov diffeomorphism is a smooth diffeomorphism of a smooth compact manifold into itself under which the tangent bundle of the manifold splits as the continuous sum of two invariant subbundles one of which contracts while the other expands under the action of the diffeomorphism. Introduced by D. V. Anosov as a straightforward analogue of the geodesic flows of Riemannian manifolds of negative curvature, these diffeomorphisms combine seemingly erratic dynamical behaviour with strong structural stability.

Since the very introduction of the subject it has been conjectured that Anosov diffeomorphisms can occur only on a particular class of homogeneous manifolds called infranilmanifolds. The conjecture is still unsettled. Noteworthy efforts are few, the most significant being arguably the work of M. Gromov on the closely allied subject of expanding maps.

The purpose of this talk will be to describe a connection which is canonically attached to an Anosov diffeomorphism on a Riemannian manifold. The connection is flat along the stable and unstable submanifolds and enjoys curious invariance properties.

Date received: June 2, 2000