Hyperbolicity of chain transitive sets with limit shadowing (Q2925712)

Let \(f\) be a diffeomorphism map on \(M\), a smooth compact manifold. We recall that a subset \(\Lambda\) of \(M\) has a hyperbolic structure property (HSP for short) if the tangent bundle restricted to \(\Lambda\) is the direct sum of an expanding \(Df\)-invariant subbundle and a contracting \(Df\)-invariant subbundle (\(Df\) denotes the differential of \(f\)) and when \(\Lambda=M\), we say that \(f\) is an \textit{Anosov diffeomorphism}. With respect to \(f\), the authors recall the following definitions: \(\Lambda\) is a \(\delta\)-trajectory, a chain transitive, a limit shadowable, a \(C^1\)-stability limit shadowable, locally maximal, and a chain transitive, respectively. Also they give the definition of an \(\varepsilon\)-shadowed sequence through a point in \(\Lambda\). Then, on using these definitions, the authors give a necessary and sufficient condition for \(\Lambda\) to have the HSP (Theorem A). Next, by considering \(\mathcal{R}\) a subset of \(\mathrm{Diff}^1(M)\), the set of \(C^1\)-diffeomorphisms on \(M\), such that \(\mathcal{R}\) is the complement of a meager set, the authors state that a locally maximal and a chain transitive \(\Lambda\) for \(f\in \mathcal{R}\) is limit shadowable if and only if \(\Lambda\) has the HSP (Theorem B).