Shadowing, internal chain transitivity and \(\alpha \)-limit sets (Q2195195)

In the present paper, some approximation theorems for shadowing dynamical systems are given. The following are the main results of the paper. Theorem 1. Let \((X,f)\) be a dynamical system with shadowing. Then, for each \(\varepsilon > 0\) and any \(A\in\mathrm{ICT}_f\) there is a full trajectory \([x_i]_{i\in Z}\) such that: 1) \(d_H(\omega(x_0),A)< \varepsilon\); 2) \(d_H(\alpha([x_i]),A)< \varepsilon\). In particular, every element of \(\mathrm{ICT}_f\) is in both \(\overline{\alpha_f}\) and \(\overline{\omega_f}\). Theorem 2. Let \((X,f)\) be a dynamical system with two-sided \(s\)-limit shadowing. Then for any \(A\in\mathrm{ICT}_f\) there is a full trajectory \([x_i]_{i\in Z}\) such that \(\alpha([x_i])=\omega([x_i])=A\). In particular, \(\alpha_f=\omega_f=\mathrm{ICT}_f\). Theorem 3. Let \((X,f)\) be a dynamical system. The following statements are equivalent: 1) \(f\) has the backward cofinal orbital shadowing property; 2) \(f\) has the backward eventual strong orbital shadowing property; 3) \(\overline{\alpha_f}=\mathrm{ICT}_f\).