For each natural number n, we consider the subgroup \(\mathcal {R}_n\) of \(\hbox {Homeo}_+[0,1]\) made by the elements that are affine except for a subset whose Cantor-Bendixson rank is less than or equal to n. These groups of generalized piecewise-affine transformations yield an ascending chain of groups as we increase n. We study how the notion of distorted element changes along this chain. Our main result establishes that for each natural number n, there exists an element that is undistorted of \(\mathcal {R}_n\) yet distorted in \(\mathcal {R}_{n+1}\) . Actually, such an element is explicitly constructed.
