Regular maps with the specification property (Q379474)

A \(C^1\) map of a smooth manifold \(M\) into itself is called regular if its derivatives are surjective. Let \({\mathcal R}(M)\) be the set of regular maps. A locally maximal \(f\)-invariant set \(\Lambda\) is called elementary if \(f|_{\Lambda}\) is topologically transitive. The authors prove the following two main results.NEWLINENEWLINE(1) Let \(\Lambda\) be a locally maximal \(f\)-invariant set for some \(f\in {\mathcal R}(M)\). Then \(f|_{\Lambda}\) has the \(C^1\) stable specification property if and only if \(\Lambda\) is a hyperbolic elementary set.NEWLINENEWLINE(2) There exists a residual subset \({\mathcal G}\) of \({\mathcal R}(M)\) with the following property: If \(\Lambda\) is a locally maximal \(f\)-invariant set for some \(f\in {\mathcal G}\), then \(f|_{\Lambda}\) has the specification property if and only if \(\Lambda\) is a hyperbolic elementary set.