Shadowable chain transitive sets of \(C^1\)-generic diffeomorphisms (Q2880466)

Let \(f\) be a diffeomorphism acting on a closed \(C^\infty\) manifold \(M\). Endow the space \(\text{Diff}(M)\) of all diffeomorphisms of \(M\) with \(C^1\)-topology.NEWLINENEWLINEA chain component \(\Lambda\subset M\) of the chain-recurrent set \(\mathrm{CR}(f)\) is said to be locally maximal if there exists an open set \(V\supset \Lambda\) such that \(\Lambda=\bigcap_{n\in \mathbb{Z}}f^n(V)\). A closed \(f\)-invariant set \(\Lambda\) is called shadowable if for every \(\varepsilon>0\) there is \(\delta>0\) such that any \(\delta\)-pseudo-orbit contained in \(\Lambda\) can be \(\varepsilon\)-traced by a point in \(M\).NEWLINENEWLINEThe main result of the paper says that there is a residual subset \(R\subset \text{Diff}(f)\) such that for every \(f\in R\) and every locally maximal chain component \(\Lambda\) of \(\mathrm{CR}(f)\) the set \(\Lambda\) is hyperbolic if and only if it is shadowable.