A note on iteration groups (Q1068966)

Answering a question of O. Borůvka and F. Neumann, \textit{G. Blanton} has found a (finitely generated) group (under composition) of \(C^{\infty}\) bijections of an open real interval I onto itself whose graphs are disjoint and dense in \(I^ 2\), which is not a subgroup of any group of \(C^ 1\) bijections of I whose graphs are disjoint and cover \(I^ 2\) [cf. C. R. Math. Rep. Acad. Sci., Soc. R. Can. 5, 169-172 (1983; Zbl 0518.26003)]. The purpose of the paper under review is ''to prove a result which is weaker than Blanton's but whose proof is much easier and shorter'': If \[ \rho (x)=\sum^{\infty}_{n=0}2^{-n} \cos (3^ n2\pi x),\quad \chi (x)=\int^{x}_{0}\exp [\rho (t)]dt \] for \(x\in {\mathbb{R}}\), \(\phi_ t(x)=\chi [\chi^{-1}(x)+t]\) for \(x\in \chi ({\mathbb{R}})\), \(t\in {\mathbb{R}}\) and T is the additive group of triadic rationals, then \(\{\phi_ t: t\in T\}\) is a group (under composition) of \(C^ 2\) bijections of the interval \(\chi({\mathbb{R}})\) onto itself whose graphs are disjoint and dense in \(\chi({\mathbb{R}})^ 2\) (and a subgroup of the group \(\{\phi_ t: t\in {\mathbb{R}}\}\) of \(C^ 1\) bijections) but it is not a subgroup of any group of \(C^ 2\) bijections of \(\chi({\mathbb{R}})\) whose graphs are disjoint and cover \(\chi ({\mathbb{R}})^ 2\).