Abstract: The ergodic theory of uniformly hyperbolic, or Axiom A,diffeomorphisms has been studied extensively, beginning with the pioneeringwork of Anosov, Sinai, Ruelle and Bowen. Much work in dynamical systems since then has been directed toward extending their results beyond Axiom A.Two basic ways to relax the condition of uniform hyperbolicity on a diffeomorphism f : M -> M are:
(a) Nonuniform hyperbolicity. Consider Lyapunov exponents and assume that none of them are zero. Most orbits are hyperbolic, but the hyperbolicityis non-uniform.
(b) Partial hyperbolicity. Permit some tangent directions on whichTf acts neutrally, but require that Tf is uniformly hyperbolic in someother tangent directions. No orbits need be completely hyperbolic, but all haveuniformly hyperbolic parts.
Approach (a) is due to Pesin. Approach (b) is due to Brin and Pesin, and is the one I will discuss. I will survey exciting new results in the study of partially hyperbolic diffeomorphisms. The theme of this research is "a little hyperbolicity goes a long way" toward ensuring robust mixing properties.