Is there a harmonic analysis theory for the group of strictly increasing continuous invertibe functions from $\mathbb{R} \mapsto \mathbb{R}$

Question

I have very little to add to the title, would appreciate pointers to the literature. A follow up question I have is: If one considers differentiable strictly increasing functions, do they lend themselves to Lie group techniques.

Answer 1

I think what you're asking about is the structure of the group $G$ of strictly increasing continuous surjections of $\mathbb R$ to itself, under the operation of composition.. The group is non-abelian and not countably generated. Moreover, the subgroup generated by "typical" (in whatever sense) elements $g_1, \ldots, g_n$ will be free.

You could consider these functions as mapping the extended reals $\overline{\mathbb R} = \mathbb R \cup \{-\infty, \infty\}$ to itself, fixing $-\infty$ and $+\infty$. Since $\overline{\mathbb R}$ is homeomorphic to a closed interval, say $I = [0,1]$, you could equivalently consider strictly increasing continuous surjections of $I$ to itself. You could use the topology of uniform convergence on $I$, making $G$ into a metrizable topological group. It is separable, but not locally compact and not complete. I don't think there can be anything resembling a Haar measure, so I don't know what kind of "harmonic analysis" you could have here.