Recurrence and the shadowing property (Q1924686)

Let \(f:X \leftarrow X\) be a homeomorphism on a compact metric space. Let \(\Omega(f)\) denote the set of nonwandering points and \(C(f)= \{x\in X: x\in \alpha (x)\cap \omega (x)\}\), where \(\alpha(x)\) and \(\omega(x)\) denote the positive and negative limit set of \(x\). The main results are: (1) If \(f\) is expansive and shadowing, then \(\overline {C(f)} (= \Omega (f))\) has a local product structure. (2) If \(f\) is shadowing and \(f|_{\Omega (f)}\) is topologically transitive, then \(X= \Omega(f)\). (3) If \(X\) is connected, is not a minimal set and \(C(f) =X\), then \(f\) is not shadowing.