A generic incompressible flow is topological mixing (Q957559)

Let \(M\) be an \(n\)-dimensional compact, connected, boundaryless \(C^{\infty}\) Riemannian manifold and \(\mu\) be the measure induced by a volume-form defined on \(M\), \(\mathfrak{X}_{\mu}^r(M)\) be for any \(r\geq 1\) the class of \(C^r\) divergence-free vector fields on \(M\). It is proved that in \(\mathfrak{X}_{\mu}^r(M)\) there exists a residual subset, such that any vector field in it has the following property: For any two nonempty open subsets \(U\) and \(V\) on \(M\) there exists \(\tau \in \mathbb{R}\) such that \(X^t(U)\bigcap V\neq \emptyset\) for any \(t\geq \tau.\) Here \(X^t\) is an associated flow with any \(C^1\) vector field \(X\), which is an \(1\)-parameter group of a \(C^1\) volume-preserving diffeomorphisms and called as incompressible flow.