The authors’ abstract: “This is a survey about the dynamical connections between a continuous self-mapping \(f\) on a compact metric space \(X\) and the shift map on the inverse limit space of the sequence \[ X@<f<< X @<f<< X @<f<< \dots\;. \] Denote the inverse limit space by \(\varprojlim (X,f)\). Let \(\sigma_ f\) denote the shift map on \(\varprojlim(X,f)\). We collect some known dynamical properties that \(f\) has these properties if and only if \(\sigma_ f\) has them. Properties covered in this survey are positive topological entropy, chaos, shadowing property and so on. We then introduce a theorem which relates the set of the chain recurrent points \(CR(f)\) of \(f\) and that of \(\sigma_ f\) (i.e. \(CR(\sigma_ g)=\varprojlim (CR(f),f))\). Similar properties are also covered for nonwandering points, recurrent points, \(\omega\)-limit points and almost periodic points.
For the entire collection see [Zbl 0777.00054].