Cone exchange transformations and boundedness of orbits

We introduce a class of two-dimensional piecewise isometries on the plane that we refer to as cone exchange transformations (CETs). These are generalizations of interval exchange transformations (IETs) to 2D unbounded domains. We show for a typical CET that boundedness of orbits is determined by ergodic properties of an associated IET and a quantity we refer to as the ‘flux at infinity’. In particular we show, under an assumption of unique ergodicity of the associated IET, that a positive flux at infinity implies unboundedness of almost all orbits outside some bounded region, while a negative flux at infinity implies boundedness of all orbits. We also discuss some examples of CETs for which the flux is zero and/or we do not have unique ergodicity of the associated IET; in these cases (which are of great interest from the point of view of applications such as dual billiards) it remains an outstanding problem to find computable necessary and sufficient conditions for boundedness of orbits.