View source for Partial hyperbolicity

==Introduction==

The theory of hyperbolic dynamical systems initially developed from the seminal notion of [[hyperbolic dynamics| uniform hyperbolicity]]. This notion can be generalized in several ways. One of these is to retain hyperbolicity without uniformity, which leads to the theory of [[nonuniform hyperbolicity|nonuniformly hyperbolic dynamical systems]], a theory that is useful in a wide range of applications. This entry describes another generalization, which retains uniformity without hyperbolicity by allowing a center direction in which any expansion or contraction is in a uniform way slower than the expansion and contraction in the unstable and stable subspaces. This theory of partially hyperbolic systems has demonstrated that a limited amount of uniform expansion and contraction is often sufficient to produce ergodicity and topological transitivity.