The specification property for flows from the robust and generic viewpoint (Q441232)

The paper studies \(C^1\)-vector fields on a smooth manifold \(M^n\), \(n\geq3\), satisfying the weak specification property robustly on an isolated compact invariant set \(\Lambda \subset M\). The authors' first main result shows that weak specification in a robust sense implies topological mixing and uniform hyperbolicity on \(\Lambda\). In this context, `robust' means that every other \(C^1\)-vector field, sufficiently \(C^1\)-close to the given one, also has an isolated invariant set with the weak specification property. Furthermore, the authors prove the generic result that the weak specification property on \(M\) (compact) implies Anosov on a residual subset of \(\mathfrak{X}^1(M)\), the space of all \(C^1\)-vector fields on \(M\) endowed with the \(C^1\)-topology. Similar results for \(C^1\)-diffeomorphisms were proved in [\textit{K. Sakai} et al., Proc. Am. Math. Soc. 138, No. 1, 315--321 (2010; Zbl 1186.37012)].