Chain recurrent sets of generic mappings on compact spaces

Abstract

Let 0-CR denote the class of all metric compacta X such that the set of maps $f:X\to X$ with 0-dimensional sets CR(f) of chain recurrent points is a dense $G_\delta$-subset of the mapping space C(X,X) (with the uniform convergence). We prove, among others, that countable products of polyhedra or locally connected curves belong to 0-CR. Compacta that admit, for each $\epsilon >0$, an $\epsilon$-retraction onto a subspace from 0-CR belong to 0-CR themselves. Perfect ANR-compacta or n-dimensional $LC^{n-1}$-compacta have perfect CR(f) for a generic self-map f. In the cases of polyhedra, compact Hilbert cube manifolds, local dendrites and their finite products, a generic f has CR(f) being a Cantor set and the set of periodic points of f of arbitrarily large periods is dense in CR(f). The results extend some known facts about CR(f) of generic self-maps f on PL-manifolds.