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:

Do such groups have a name?

What is the classification of all such groups? (with the properties of $f'(x) >0$ and $f\in C^3$ if possible)

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.

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