Let us denote by the symbol $\mathcal{G}$, a group of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ (with the composition operation) that is additionally closed under all affine change of variables of the form (homothety):


$$ h(x) = mx, m>0$$


In other words, I would like the following property to hold for any affine maps $h$ (of the above form):
$$hGh^{-1} \in G$$


Intuitively such a group is a group of functions that is invariant under all rescalings.


A simple example of such a group is the group of fractional linear transformations (FLT) (with real coefficients), namely the group $\mathcal{S}$ consisting of that $f(x) = \frac{ax+b}{cx+d}$ where $a,b,c,d \in R$.


My questions are:


1. Do such groups have a name?
2. What is the classification of all such groups? (with the properties of $f'(x) >0$ and $f\in C^3$ if possible)
3. Is there a general way of constructing such groups or putting this question in a general context?


Thank you in advance to all those who respond,


E(up)lio M.


.



What classes of functions are closed under all rescalings?
If $G$ contains the linear group consisting of all functions of the form $h\_m(x) = mx$ then obviously it satisfies your conditions.


Conversely, by the definition of the derivative,


(\*) $lim\_{m \to \infty} (h\_m^{-1} f h\_m(x)) = f'(0) x$


So the closure $\bar G$ of your group (in some appropriate topology) contains all linear functions of the form $h\_m$ where $m = f'(0)$ for $f \in G$. So if you add to your conditions that the group $G$ be closed, and that the set of derivative values $f'(0)$, $f \in G$ consists of all positive real numbers, then $G$ does indeed contain the linear group.


