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.