ABSTRACT:

We provide a general mechanism for obtaining uniform information from pointwise data even when the pertinent quantities are highly discontinuous. Some of the applications are: If a diffeomorphism of a compact Riemannian manifold has nonzero Lyapunov exponents everywhere then the nonwandering set iis uniformly hyperbolic. If, in addition, there are expanding and contracting invariant cone families, which need not be continuous, then the diffeomorphism is an Anosov diffeomorphism, i.e. the entire manifold is uniformly hyperbolic. There are also several applications to partially hyperbolic systems and dominated splitting.