An engine breaking the \(\Omega\)EP-property (Q860497)

Let \(f:X\rightarrow X\) be a map of a continuum (i.e., a compact, connected metric space) to itself. By \(\Omega(f)\), the nonwandering points of \(f\), we mean those \(x\in X\) such that for any open neighborhood \(U\) of \(x\), there exists \(y\in U\) and \(n\in\mathbb N\) such that \(f^n(y)\in U\). We call \(x\) an eventually periodic point if for some \(m\in\mathbb N\), \(f^m(x)\) is a periodic point of \(f\). The set of such \(x\) is denoted \(EP(f)\). One says that \(X\) has the nonwandering-eventually-periodic property, abbreviated \(\Omega\)EP-property, if for every map \(f:X\rightarrow X\), \(\Omega(f)\subset\)cl\((EP(f))\). In Section 2, the authors point out that arcs and trees have the \(\Omega\)EP-property but that neither the null-comb [\textit{J. Charatonik} and \textit{A. Illanes}, Topology Appl. 144, 109--132 (2004; Zbl 1057.54025)] nor the topologist's sine curve has the \(\Omega\)EP-property. In Sections 4 and 5, the authors construct respectively the ``basic function'' \(\varphi:[0,1]\rightarrow[0,1]\) and the ``advanced function'' \(\psi:[0,1]\rightarrow[0,1]\) by modifying \(\varphi\). The latter is used to prove that neither the harmonic fan (cone over the convergent sequence) nor the topologist's sine curve has the \(\Omega\)EP-property. In Section 5 they prove: Theorem 3: Let \(X\) be a nonlocally connected continuum with a point \(p\in X\) such that \(X\) is not connected \textit{im kleinen} at \(p\). If there is an arc \(A\) in \(X\) containing \(p\), then the continuum \(X\) does not have the \(\Omega\)EP-property. Section 9 ends the paper with several questions.