Distortion in the group of locally monotone homeomorphisms of a Cantor set and in the group of generalized interval exchange transformations

Let f be either a generalized interval exchange transformation or a locally monotone homeomorphism of a Cantor subset of the real line. In this article, we prove that the following are equivalent. 1. The number of discontinuities of f^n is bounded. 2. There exists n $\ge$ 1 such that the element f is conjugate to the restriction to a closed invariant subset of a disjoint union of n circles of a homeomorphism of this disjoint union of circles. 3. The element f is distorted in the group of generalized interval exchange transformations or in the group of locally monotone homeomorphisms of the Cantor subset.