Partially hyperbolic sets with a dynamically minimal lamination (Q1661089)

Let \(f\) be a \(C^1\) diffeomorphism of a closed smooth manifold and let \(\Lambda\) be a partially hyperbolic transitive invariant set of \(f\) with bundles \(E^s,E^c,E^u\) (where it is assumed that any of the bundles is nontrivial). Let \(F^s\) and \(F^u\) be the laminations which integrate \(E^s\) and \(E^u\), respectively. The set \(\Lambda\) is called weak \(s\)-minimal (weak \(u\)-minimal) if the orbit of any leaf in \(F^s\) (in \(F^u\)) has a dense intersection with \(\Lambda\). The author shows that if \(\Lambda\) is a proper subset of the manifold which is weak \(s\)-minimal (or weak \(u\)-minimal), then its interior is empty.