On the Plaque Expansivity Conjecture

It is one of the main properties of uniformly hyperbolic dynamics that points of two distinct trajectories cannot be uniformly close one to another. This characteristics of hyperbolic dynamics is called expansivity. Hirsch, Pugh and Shub, 1977, formulated the so-called Plaque Expansivity Conjecture, assuming that two invariant sequences of leaves of central manifolds, corresponding to a partially hyperbolic diffeomorphism, cannot be locally close. There are many important statements in the theory of partial hyperbolicity that can be proved provided Plaque Expansivity Conjecture holds true. Here we are proving this conjecture in its general form.Comment: The proof written here was wrong. I hope to replace this with a correct on

Sets of invariant measures and Cesaro stability

Sets of invariant measures are considered for continuous maps of a metric compact set. We take Kantorovich metric to calculate distance between measures and Hausdorff metrics to calculate distance between compact sets. Consider the function that makes correspondence between a continuous map and the set of all its Borel probability invariant measures. We demonstrate that a typical map is a continuity point of that function. Using approaches of Takens' tolerance stability theory we provide some corollaries that demonstrate that for a typical map points are structurally stable in a statistical sense.Comment: 11 pages, no figure