Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms (Q2884390)

In this paper the authors study generic volume-preserving diffeomorphisms on compact manifolds. Let \(\text{Diff}^r_m(M)\) be the set of \(m\)-preserving \(C^r\)-diffeomorphisms endowed with the \(C^r\)-topology \((r \geq 1)\), where \(M\) is a smooth closed manifold \((\dim M \geq 2)\) and \(m\) is a smooth volume measure. For \(f \in \text{Diff}^1_m(M)\), a point \(x\) (or its orbit) is called nonuniformly hyperbolic if all its Lyapunov exponents are nonzero and the set of those points is denoted by \(\text{Nuh}(f)\). It is shown that there is a residual set \(\mathcal{R} \subset \text{Diff}^1_m(M)\) such that for every \(f \in \mathcal{R}\), either \(m(\text{Nuh}(f)) = 0\) or the restriction \(f|\text{Nuh}(f)\) is ergodic and the orbit of almost every point in \(\text{Nuh}(f)\) is dense in \(M\). As a corollary, we have that there is a residual set \(\mathcal{R} \subset \text{Diff}^1_m(M)\) such that for every \(f \in \mathcal{R}\), either \(m(\text{Nuh}(f)) = 0\) or there is a global dominated splitting.NEWLINENEWLINEFor the proof, the authors establish some new properties of independent interest that hold \(C^r\)-generically for any \(r \geq 1\); namely, the continuity of the ergodic decomposition, the persistence of invariant sets, and the \(L^1\)-continuity of Lyapunov exponents.