Let $X,Y$ be two topological spaces. One can define various topologies on $C(X,Y),$ the space of continuous functions from $X$ to $Y.$ Let us take the group $G=Hom(X),$ the homeomorphisms of $X.$ Let $G_1$ be a subgroup of $G.$
Now, given $G_1$ and a $f\in C(X,Y),$ I am looking for "useful" nontrivial topologies $\tau$ on $C(X,Y)$ so that the set $\left\{f\circ h\colon h\in G_1\right\}$ has some nice properties with respect to $\tau.$
For example, when $X=Y=\mathbb{R}$ and $G_1$ is the group ,say, generated by $x\mapsto 2x,$ I want to say that $\left\{f(x),f(2x),f(4x),\ldots; f(x/2),\ldots\right\}$ satisfy a topological property. Or, if $G_1=F$ and $f\colon I\rightarrow X$ then $F$ operates on $C(I,X),$ so , what I want is a topology $\tau$ on $C(X,Y)$ such that the embedding $F\hookrightarrow Hom(C(I,X))$ has nice properties. Thank you.