On a problem of Akin and Carlson on dense orbits and transitivity (Q1709068)

A (discrete) dynamical system is a pair \((X, f)\) where \(X\) is a topological space and \(f\) is a continuous self-map on \(X\). Moreover, a normal Hausdorff space is called \(T_5\)-space (or hereditarily normal) if every subspace is normal and \(T_6\)-space (or perfectly normal) if every closed subset is a \(G_{\delta}\)-set. In [Topol. Appl. 159, 2815--2830 (2012; Zbl 1248.37015)] \textit{E. Akin} et al. raised the following question: does there exist a transitive self-map on a separable compact Hausdorff space which has no dense orbits? The aim of this paper is to give an answer to this question for the class of perfectly normal spaces. The authors consider the following cases: 1) A separable Hausdorff Baire space admitting a transitive homeomorphism without dense orbits. 2) A separable Hausdorff Baire space admitting a mixing map without dense orbits. The following results are provided: 1) Let \(f\) be a transitive self-map on a \(T_3\)-space \(X\) with a countable dense set \(D\) of \(G_{\delta}\)-points. If \(X\) is countably compact, then \(f\) has a dense orbit. 2) Let \(f\) be a transitive self-map on a s-separable countably compact \(T_3\)-space \(X\) such that \(X\setminus \{x\}\) is normal for every \(x \in X\). Then \(f\) has a dense orbit. In particular they obtain a negative answer to the problem of \textit{E. Akin} and \textit{J. D. Carlson} [Topology Appl. 159, No. 12, 2815--2830 (2012; Zbl 1248.37015)] in the realm of \(T_6\)-spaces. 3) Let \(f\) be a transitive self-map on a separable countably compact \(T_6\)-space \(X\). Then \(f\) has a dense orbit.