Finite generating sets for monoids of $G$-equivariant functions

Given the action of a group $G$ on a set $X$, the set of all $G$-equivariant functions, i.e., those satisfying $f(g\cdot x)=g\cdot f(x)$ for all $g\in G$ and $x\in X$, forms a monoid under composition. In this work we study their generating sets. First, we propose bounds for the cardinalities of the generating sets of their group of units, denoted by $\operatorname{Aut}_{G}(X)$. Subsequently, using so-called orbital infiltrations, certain transformations that turn out to be indispensable and provide relevant structural information about the monoid, we determine conditions on the group $G$, the set $X$, and the action that prevent the whole monoid $\operatorname{End}_{G}(X)$ from admitting a finite generating set.