Uniform simplicity for subgroups of piecewise continuous bijections of the unit interval

Let $I=[0,1)$ and $\mathcal{PC}(I)$ [resp. $\mathcal{PC}^+(I)$] be the quotient group of the group of all piecewise continuous [resp. piecewise continuous and orientation preserving] bijections of $I$ by its normal subgroup consisting in elements with finite support (i.e. that are trivial except at possibly finitely many points). Unpublished Theorems of Arnoux ([Arn81b]) state that $\mathcal{PC}^+(I)$ and certain groups of interval exchanges are simple, their proofs are the purpose of the Appendix. Dealing with piecewise direct affine maps, we prove the simplicity of the group $\mathcal A^+(I)$ (see Definition 1.6). These results can be improved. Indeed, a group $G$ is uniformly simple if there exists a positive integer $N$ such that for any $f,\phi \in G\setminus\{Id\}$, the element $\phi$ can be written as a product of at most $N$ conjugates of $f$ or $f^{-1}$. We provide conditions which guarantee that a subgroup $G$ of $\mathcal{PC}(I)$ is uniformly simple. As Corollaries, we obtain that $\mathcal{PC}(I)$, $\mathcal{PC}^+(I)$, $PL^+ (\mathbb S^1)$, $\mathcal A(I)$, $\mathcal A^+(I)$ and some Thompson like groups included the Thompson group $T$ are uniformly simple.