Construction of Invariant Curves for Some Piecewise Isometries

We establish conditions for the existence of a family of piecewise linear invariant curves in a two-parameter family of piecewise isometries on the upper half-plane known as Translated Cone Exchange Transformations. We show that these curves are embeddings of interval exchange transformations and give rise to layers of invariant regions. We also show the existence of a trapezoidal piecewise isometry for which the dynamics on the top and bottom edges are distinct 2-interval exchange transformations.