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A relaxation of the notion of invariant set, known as $k$-quasi-invariantset, has appeared several times in the literature in relation to groupdynamics. The results obtained in this context depend on the fact that thedynamic is generated by a group. In our work, we consider the notions ofinvariant and 1-internally-quasi-invariant sets as applied to an action of afunction $f$ 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 subsetof $I$ 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 whichevery finite subset of $I$ admits a finite internally-$k$-quasi-invariantsuperset? This parallels a similar investigation undertaken by C. E. Praeger inthe context of group actions.

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