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 problems of the following type, where k ∈ \0,1\: what are the functions f for which every finite subset of I is internally-k-quasi-invariant? More restrictively, if I = 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 superset that is internally-k-quasi-invariant? This parallels a similar investigation undergone by C. E. Praeger in the context of group actions.