On the Plaque Expansivity Conjecture

TL;DR

This paper proves the Plaque Expansivity Conjecture for partially hyperbolic diffeomorphisms, establishing a key property of hyperbolic dynamics that invariant leaves cannot be locally close, which supports many results in the field.

Contribution

It provides a proof of the Plaque Expansivity Conjecture in its general form, a longstanding open problem in the theory of partial hyperbolicity.

Findings

Proves the Plaque Expansivity Conjecture for general cases

Confirms that invariant leaves of central manifolds cannot be locally close

Supports the validity of many results in partial hyperbolic dynamics

Abstract

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.