Chain recurrent sets of generic mappings on compact spaces (Q260572)

The space of maps \(f : X \rightarrow X\) with the topology of uniform convergence is denoted by \(C(X, X)\). Suppose that the subset of all maps in \(C(X, X)\) satisfying some property \(\mathcal P\) contains a dense \(G_{\delta}\) subset; then we call such a map with the property \(\mathcal P\) generic. NEWLINENEWLINEThe authors investigate the chain recurrent sets \(CR(f)\) of generic maps \(f : X \rightarrow X\) on some compacta.NEWLINENEWLINELet 0-\(CR\) denote the class of all metric compacta \(X\) such that for a generic \(f : X \rightarrow X\) the set \(CR(f)\) is 0-dimensional. Amongst others, the following results are shown:NEWLINENEWLINECountable products of polyhedra or locally connected curves belong to 0-\(CR\).NEWLINENEWLINECompacta that admit, for each \({\epsilon}> 0\), an \({\epsilon}\)-retraction onto a subspace that is contained in 0-\(CR\) belong to 0-\(CR\) themselves.NEWLINENEWLINEFor a generic \(f\), \(CR(f)\) is homeomorphic to the Cantor set and the set of periodic points of \(f\) of arbitrarily large periods is dense in \(CR(f)\) if \(f\) is defined on polyhedra, compact Hilbert cube manifolds, local dendrites and finite products of them.