The local star condition for generic transitive diffeomorphisms (Q2808605)

Let \(M\) be a closed \(C^{\infty}\)-manifold, and \(\mathrm{Diff}(M)\) be the set of diffeomorphisms defined on \(M\) with the \(C^1\)-topology. An invariant set is said to be hyperbolic for a diffeomorphism \(f\) if the tangent bundle has a continuous splitting into two sub-bundles with respect to the derivative operator of \(f\), and the derivative operator is uniformly contracting or expanding along these two sub-bundles. R. Mañé raised a question that if \(f\) satisfies the star condition, then \(f\) is Axiom A, where the star condition means that there is a \(C^1\)-neighborhood of \(f\) such that any diffeomorphism in this neighborhood has only hyperbolic periodic points; the Axiom A refers to the requirements that the non-wandering set is hyperbolic and it is the closure of the periodic points [\textit{R. Mañé}, Topology 17, 383--396 (1978; Zbl 0405.58035)]. The author uses a stronger assumption, the local star condition, to study Mañé's problem. A diffeomorphism \(f\) satisfies the local star condition is that there is a \(C^1\)-neighborhood of \(f\) such that for any diffeomorphism in this neighborhood, the periodic points in the locally maximal set is hyperbolic, where a set is locally maximal if there is a neighborhood \(U\) of this set such that this set is equal to \(\cap_{n\in\mathbb{Z}}f^n(U)\).NEWLINENEWLINEThe author shows that there is a set, which is the intersection of a countable family of open and dense subsets of \(\mathrm{Diff}(M)\), for any diffeomorphism in this set, if \(f\) satisfies the local star condition on a transitive set, then it is hyperbolic. The whole argument is dependent on the properties of the assumption on the locally maximal and transitive set, the dominated splitting, and ergodic measures.