45 research outputs found
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