A note on shadowing with chain transitivity (Q434828)

The author analyzes how shadowing, chain transitivity, and other similar dynamical concepts transfer from a map \(f: X\mapsto X\) on a compact metric space \(X\) to the induced maps \(f_K\) on the hyperspace of \(X\) and \(f_M\) on the space of probability measures on \(X\). The results presented here include:NEWLINENEWLINE(1) shadowing of \(f\) and chain transitivity of \(f\times f\) implies chain mixing of every iteration of \(f_K\);NEWLINENEWLINE(2) topological ergodicity of \(f\times f\) implies topological ergodicity of \(f_M\times f_M\);NEWLINENEWLINE(3) shadowing and chain transitivity of \(f\) imply that either every iteration of \(f\) is equicontinuous, or every iteration of \(f\) is syndetically sensitive.NEWLINENEWLINENote: in the proof of (1) the author claims that shadowing of \(f\) easily implies shadowing of \(f_K\). This seems to be true, but I do not see a fast elementary proof.