Abstract
We provide a general mechanism for obtaining uniform information from pointwise data. For instance, a diffeomorphism of a compact Riemannian manifold with pointwise expanding and contracting continuous invariant cone families is an Anosov diffeomorphism, i.e., the entire manifold is uniformly hyperbolic.
Highlights
We present a novel combination of ideas that provides a way of obtaining uniform information from nonuniform assuptions
Dx f n(v ) ≥ A(x)λn(x) v for every n ∈ N and x ∈ X, whenever v belongs to a certain subspace Ex of Tx M
This condition can be characterized by the existence of an invariant cone family C such that vectors in C (x) expand in the same way as above
Summary
We present a novel combination of ideas (from descriptive set theory and hyperbolic dynamical systems) that provides a way of obtaining uniform information from nonuniform assuptions. This condition can be characterized by the existence of an invariant cone family C such that vectors in C (x) expand in the same way as above. We stress that our approach is quite different and allows us to extend our result to the case when the cone family is not continuous. In [7], Mañé proved a statement similar to ours that differs in two ways On one hand, he does not require the expansion to be exponential (just unboundedness of orbits of the differential). Descriptive set theory has been used for the study of topological dynamical systems, but we have not seen it applied to smooth dynamics. A selection of pertinent references is [1, 3, 4, 5, 7]
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