The essential coexistence phenomenon in dynamics

Abstract

This article is a survey of recent results on the essential coexistence of hyperbolic and non-hyperbolic behaviour in dynamics. Though in the absence of a general theory, the coexistence phenomenon has been shown in various systems during the last three decades. We will describe the contemporary state of the art in this area with emphasis on some new examples in smooth conservative systems, in both cases of discrete and continuous-time.

Keywords:

• pointwise partial hyperbolicity
• Lyapunov exponents
• ergodicity
• accessibility

Acknowledgements

J. Chen and Ya. Pesin were partially supported by NSF grant DMS-1101165.

Notes

1. In many interesting examples the set is open, see below.

2. Note that since the Lyapunov exponents at almost every are all nonzero, the map has at most countably many ergodic components of positive measure (see [6,7]).

3. We say that a foliation W on is absolutely continuous if for almost every there is a neighbourhood B(x, q(x)) such that for almost every y ∈ B(x, q(x)) the conditional measure generated on the local leaves V(y) by volume m is absolutely continuous with respect to the leaf volume m _V(y) on V(y).

4. A diffeomorphism f that is pointwise partially hyperbolic on an open set is called dynamically coherent if the subbundles E  ^cu = E  ^c ⊕ E  ^u , E  ^c , and E  ^cs = E  ^c ⊕ E  ^s are integrable to continuous foliations with smooth leaves W  ^cu , W  ^c and W  ^cs , called respectively the centre-unstable, centre and centre-stable foliations. Furthermore, the foliations W  ^c and W  ^u are subfoliations of W  ^cu , while W  ^c and W  ^s are subfoliations of W  ^cs .

5. We stress that V  ⁱ (z _k−1) is the local leaf of W  ⁱ at z _k−1. In particular, the length of the curve γ _k (the leg of the path) does not exceed the size of V ⁱ (z _k−1).

6. In the case when f is uniformly partially hyperbolic on the whole manifold , has positive central exponents and the accessibility property, the same result as in Theorem 2.4 was proved in [33].