Functions with a maximal number of finite invariant or internally-1-quasi-invariant sets or supersets

A relaxation of the notion of invariant set, known as \(k\)-quasi-invariant set, has appeared several times in the literature in relation to group dynamics. The results obtained in this context depend on the fact that the dynamic is generated by a group. In our work, we consider the notions of invariant and 1-internally-quasi-invariant sets as applied to an action of a function \(f\) on a set \(I\). We answer several questions of the following type, where \(k \in \{0,1\}\): what are the functions \(f\) for which every finite subset of \(I\) is internally-\(k\)-quasi-invariant? More restrictively, if \(I = \mathbb{N}\), what are the functions \(f\) for which every finite interval of \(I\) is internally-\(k\)-quasi-invariant? Last, what are the functions \(f\) for which every finite subset of \(I\) admits a finite internally-\(k\)-quasi-invariant superset? This parallels a similar investigation undertaken by C. E. Praeger in the context of group actions.