Stable minimality of expanding foliations (Q2665536)

Let \(f \in \text{Diff}_m^{\, r}(M)\) be a diffeomorphism on a closed manifold \(M\) and \(W\) be a foliation of \(M\) that is invariant and uniformly expanded under \(f\). If \(W\) is a minimal foliation, then \(f\) is topologically transitive and topological mixing. The authors obtain a sufficient condition for an expanding foliation to be stably minimal. More precisely, for a \(C^1\)-generic \(f \in \text{Diff}^{\, r}_m(M)\), if it admits a minimal expanding foliation \(W\) and a hyperbolic periodic point \(p\) with \(\dim E^u_p = \dim W\), then there exists a \(C^1\)-neighborhood \(\mathcal{U}\subset \text{Diff}^{\, r}_m(M)\) of \(f\) such that for every \(g\in \mathcal{U}\), the continuation \(W_g\) is minimal. Moreover, there exists a hyperbolic periodic point \(q_g\) whose Pesin homoclinc class \(\text{Phc}_g(q_g)\) is essentially dense over \(M\). The authors also discuss several conjectures related to minimal expanding foliations.