\(C^r\)-density of (non-uniform) hyperbolicity in partially hyperbolic symplectic diffeomorphisms (Q290762)

Let \(f\) be a partially hyperbolic symplectic diffeomorphism of class \(C^r\), \(r\geq 2\), of a compact manifold \(M\) with a 2-dimensional center bundle. Let \(W^s\) and \(W^u\) be the corresponding stable and unstable foliations.NEWLINENEWLINEIt is assumed that \(f\) is accessible, i.e., for any points \(x,y\in M\) there is a path connecting \(x\) and \(y\) and consisting of finitely many subpaths each of which belongs to a leaf in \(W^s\) or \(W^u\).NEWLINENEWLINEIt is also assumed that certain pinching and bunching conditions are satisfied (these conditions are formulated in terms of expansion and contraction rates of the partially hyperbolic structure and are \(C^1\)-open).NEWLINENEWLINEFinally, it is assumed that \(f\) has a periodic point.NEWLINENEWLINEIt is shown that under the above formulated conditions, \(f\) can be \(C^r\) approximated by nonuniformly hyperbolic symplectic diffeomorphisms.