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

On local fixed or periodic point properties

“…Definition 1. A topological space X has the local fixed point property (X ∈ LF P P ) if X has an open basis B (a basis for LF P P ) such that B ∈ F P P for each B ∈ B. X has the weak local fixed point property (X ∈ wLF P P ) if X has an open basis B (a basis for wLF P P ) such that, for each B ∈ B and each continuous map f : X → X, whenever f (B) ⊂ B, then f has a fixed point in B [6].…”

“…X has the weak local periodic point property (X ∈ wLP P P ) if X has an open basis B (a basis for wLP P P ) such that, for each B ∈ B and each continuous map f : X → X, whenever f (B) ⊂ B, then f has a periodic point in B [6].…”

“…The main motivation for studying local fixed or periodic point properties is that they play an essential role in the dynamics of continuous maps: if a perfect ANRcompactum X has the wLP P P , then the set of periodic points of a generic map f : X → X has no isolated points and is dense in the set of chain recurrent points of f [6].…”

“…In particular, all polyhedra and all Euclidean manifolds have the LF P P . A harmonic fan has the LP P P but it does not have the wLF P P [6]. It is natural to ask for what continua properties LF P P , LP P P and their weak versions differ.…”

“…Next we present an example of a plane continuum X ∈ wLP P P \ wLF P P having an open basis B such that, for each map f : X → X and each B ∈ B, f has a periodic point of period at most two in B. It was observed in [6] that a harmonic fan F is a plane continuum in LP P P \ wLF P P but it is easy to see that in F it is not possible to find a bound for the periods, in the described sense.…”