A generalisation of the Babbage functional equation

A recent refinement of Ker'ekj'art'o’s Theorem has shown that in $\mathbb R$and$\mathbb R^2$all$\mathcal C^l$-solutionsofthefunctionalequation$f^n =$\text{Id}$$are$\mathcal C^l$-linearizable,where$l\in {0,1,\dots \infty}$\mathrm{.}When$l\geq 1$, in the real line we prove that the same result holds for solutions of $f^n=f$, while we can only get a local version of it in the plane. Through examples, we show that these results are no longer true when $l=0$ or when considering the functional equation $f^n=f^k$ with $n>k\ge 2$.