Genericity of nonuniform hyperbolicity in dimension 3

Abstract

For a generic conservative diffeomorphism of a closed connected 3-manifold M , the Oseledets splitting is a globally dominated splitting. Moreover, either all Lyapunov exponents vanish almost everywhere, or else the system is non-uniformly hyperbolic and ergodic. This is the 3-dimensional version of the well-known result by 4], stating that a generic conservative surface diffeomorphism is either Anosov or all Lyapunov exponents vanish almost everywhere. This result inspired and answers in the positive in dimension 3 a conjecture by Avila-Bochi [2].

New examples of stably ergodic diffeomorphisms in dimension 3

“…This follows from lemma 6.1 in [Her12]. In particular, the set Γ( f n ) is su-saturated for any n ∈ N.…”

“…Theorem 2.4. [ Her12] Let M be a closed connected manifold of dimension 3, then there exists R ⊂ Diff 1 m (M) a residual set such that every f ∈ R satisfies one of the following alternatives:…”

Symplectomorphisms with positive metric entropy

“…In the case of one‐dimensional center, a related result was proved earlier by J. Rodriguez–Hertz [17]: the $$C^r$$‐generic volume‐preserving diffeomorphism has no proper partially hyperbolic invariant compact set with one‐dimensional center and positive Lebesgue measure.…”

What are Lyapunov exponents, and why are they interesting?

“…This result was proved in dimension 2 by Mañé-Bochi [51,17] and dimension 3 by M.A. Rodriguez-Hertz [58]. Positive entropy is an a priori weak form of chaotic behavior that can be confined to an invariant set of very small measure, with trivial dynamics on the rest of the manifold.…”