The ergodic theory of uniformly hyperbolic, or Axiom A, diffeomorphisms has been studied extensively, beginning with the pioneering work of Anosov, Sinai, Ruelle and Bowen. Consider Lyapunov exponents and assume that none of them are zero.
Most orbits are hyperbolic, but the hyperbolicity is non-uniform.
Permit some tangent directions on which Tf acts neutrally, but require that Tf is uniformly hyperbolic in some other tangent directions. No orbits need be completely hyperbolic, but all have uniformly hyperbolic parts. I will survey exciting new results in the study of partially hyperbolic diffeomorphisms.