Suppose I have continuous group (or Lie group) formed by a smooth transformation $T_{a}$ (not necessarily linear or matrix-like) acting on a function $f(x,y)$ that maps to some other set of functions (or just a mapping between general function spaces which are not necessarily complete) for $x,y \in [a,b] \times [c,d].$
Are there any group-like theorems that would let me test or determine whether another given function $g(x)$ is in the orbit of the action of $T_{a}[f(x)]$?