Virtually stable maps and their fixed point sets (Q837586)

Let \(X\) be a Hausdorff topological space and \(f: X \to X\) a continuous map whose fixed point set \(F(f)\) is nonempty. A point \(x \in F(f)\) is called virtually stable if for each of its neighborhoods \(U\) there exist a neighborhood \(V\) of \(x\) and an increasing sequence \(\{k_n\}\) of positive integers such that \(f^{k_n}(V) \subseteq U\) for all \(n \in \mathbb{N}.\) A map \(f\) is called virtually stable if all of its fixed points are virtually stable. The authors study topological properties of the convergence set \(C(f) = \{x \in X\): the sequence \(\{f^n(x)\}\}\) is convergent. In particular, it is shown that if \(X\) is regular and \(f\) is virtually stable, then \(F(f)\) is a retract of \(C(f).\) It is demonstrated also that under some additional assumption of generalized homogeneity of a virtually stable map \(f\) the set \(F(f)\) is star-convex.