Diffeomorphisms with \(C^1\)-stably average shadowing (Q1942650)

The authors consider the notion of ``\(C^1\)-stably average shadowing property'' for \(C^1\)-diffeomorphisms \(f\) on a closed \(C^\infty\) manifold \(M\). The main result states that if \(f\) has the \(C^1\)-stably average shadowing property on a transitive set \({\Lambda\subset M}\), which is not a periodic orbit, then \(\Lambda\) admits a dominated splitting. The proof is based on the perturbation along the periodic orbits and the approximation for \(\Lambda\) by periodic orbits of some diffeomorphisms \(g_n\) converging to \(f\) in the \(C^1\) topology .