Existence of non-trivial embeddings of interval exchange transformations into piecewise isometries

Abstract We prove that almost every interval exchange transformation, with an associated translation surface of genus $g\geq 2$ , can be non-trivially and isometrically embedded in a family of piecewise isometries. In particular, this proves the existence of invariant curves for piecewise isometries, reminiscent of Kolmogorov–Arnold–Moser (KAM) curves for area-preserving maps, which are not unions of circle arcs or line segments.