Generic diffeomorphisms with robustly transitive sets (Q2851111)

Let \(f\) be a diffeomorphism. A set \(\Lambda\) is \textit{robustly transitive} if there is a neighborhood \(U\) of \(\Lambda\) and a \(C^{1}\)-neighborhood \(\mathcal{U}(f)\) of \(f\) such that \(\Lambda\) is locally maximal, \(\Lambda_{g}(U)=\cap_{n\in\mathbb{Z}}g^{n}(U)\) is transitive for any \(g\in\mathcal{U}(f)\). Robustly transitivity is an important research content of the global dynamics.NEWLINENEWLINEIn this paper, the authors give a condition \((P):\) Let \(\Lambda\) be a closed invariant set, for any hyperbolic periodic points \(p,q\in\Lambda,\) NEWLINE\[NEWLINE W^{s}(p)\cap W^{u}(q)\neq0, \;\;W^{u}(p)\cap W^{s}(q)\neq0. NEWLINE\]NEWLINE The main result is that when \(\Lambda\) is a robustly transitive set, for \(C^{1}\) generic \(f\), if \(f\) satisfies condition \((P)\), then \(\Lambda\) is hyperbolic.