Distortion in groups of generalized piecewise-linear transformations

For each natural number n, we consider the subgroup Rₙ of Homeo₊[0,1] made by the elements that are linear except for a subset whose Cantor-Bendixson rank is less than or equal to n. These groups of generalized piecewise-linear 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 exits an element that is undistorted of Rₙ yet distorted in Rₙ₊₁. Actually, such an element is explicitly constructed.