\(C^1\)-stably weakly shadowing homoclinic classes admit dominated splittings (Q968743)

The aim of the present paper is to introduce the notino of J\(C^1\)-stably weakly shadowing for a closed \(f\)-invariant set where \(f\) is a diffeomorphism defined in a closed \(n\)-dimensional \(C^\infty\) manifold. If \(p\) is a hyperbolic saddle periodic point of \(f\) is proved that the homoclinic class \(H_f(p)\) of \(p\), if \(f|_{H_f(p)}\) is \(C^1\)-stably the splitting on \(H_f(p)\) is partially hyperbolic and if in additon, \(f\) is far from homoclinic tagency, then \(H_f(p)\) is strongly partially hyperbolic.