On the concept of hyperbolicity

I will consider diffeomorphisms of a manifold M (discrete dynamical systems).A hyperbolic structure of an f-invariant subset N of M is a Df-invariantsplitting of the tangent bundle of N into a direct sum of contracting andexpanding subbundles. The nature of the hyperbolic structure is closely relatedto the structure of the invariant set, and the (structural) stabilityproperties of the invariant set. In particular, these well known facts have ledto a topological classification of hyperbolic attractors. A far more complexsituation is when the hyperbolic properties break down on parts of thef-invariant set, that is, the invariant set may be considered as havinghyperbolic properties on (large) parts of the invariant set, but noteverywhere. We may call such sets almost hyperbolic. In ``real world models''such behavior seems to be rather common.

I will briefly review some of the classical theory for hyperbolic invariant sets, and then turn my attention to almost hyperbolic sets. The discussion here will have to aspects; first I will consider some properties associated with global bifurcations in smooth families of diffeomorphism on (low-dimensional) manifolds. Then I will consider the ideas behind the techniques in proving the existence of ``strange attractors'' for a set of positive measure (in the parameter space) in families of dissipative diffeormorphism (still living on low-dimensional manifolds).

Keywords: Discrete dynamical systems, hyperbolic structures, global bifurcations, invariant manifolds, strange attractors and chaos.