Dynamics on locally compact Hausdorff spaces (Q2800408)

The authors study the following problem: Let \(X\) be a topological space and \(S\subset X\). When is there a continuous map \(f:X\to X\) such that \(S\) equals the set \(P(f)\) of periodic points of \(f\)? A first result reads as follows: Let \(X\) be compact and assume that \(S=P(f)\) is discrete and infinite. Then the closure \(\bar{S}\) of \(S\) is uncountable. Turning to the locally compact case the authors restrict themselves to the case where \(X\) is homeomorphic to \(\omega^2\). Considering first homeomorphisms the authors show that every set of natural numbers occurs as the set of periods for a self-homeomorphism of \(X\) and any compact subset \(K\) of \(X\) may occur as \(P(h)\) for a self-homeomorphism of \(X\), whereas its complement \(K^c\) cannot occur as some \(P(h)\) provided \(K\) is infinite. Considering finally the case of continuous self-maps the authors prove that a subset \(S\subset X\) can occur as \(P(f)\) if and only if \(\bar{S}\setminus S\) is either empty or discrete.