XXV Workshop on Geometric Methods in Physics 2-8.07.2006

Dmitry Anosov

Hyperbolicity in the Dynamical Systems Theory

In the talk a survey of the hystory and the current state of the concept of hyperbolicity in the dynamical dystems theory. Historically it goes back to the Poincare's discovery of the homoclinic points and to Hadamard"s work on the negative curvature surfaces. (In the ``technical'' respect another Hadamard's work was important - the paper on the invariant manifolds of the saddle type fixpoint.) Being not completely cognizable, these concepts still became apparent during several decades, untill they were explicitly revealed soon after the appearance (I dare even say, advent) of the Smale's horseshue. In the first version of the ``hyperbolic'' character only the most strongly pronounced hyperbolisity was considered - it was ``complete'' and ``uniform''. The contemporary stage is concerned with the weakening og the related conditions in various directions. It is closely connected to the bifurcations theory.