Interval exchange transformations and foliations on infinite genus 2-manifolds

Abstract

For each of the following properties, there is an isometric generalized interval exchange transformation (i.e. isometric GIET) having such property: (a) non-trivial recurrence orbits are exceptional and the union of them is a dense set, moreover the intersection of the closure of two such orbits is the union of finite orbits; (b) coexistence of dense orbits and exceptional orbits; (c) existence of a dense sequence of exceptional orbits $\{\mathcal{O}(p_k)\colon k=1,2,\dotsc\}$ such that $\overline{\mathcal{O}(p_1)}\subsetneqq\overline{\mathcal{O}(p_2)}\subsetneqq\dotsb\subsetneqq\overline{\mathcal{O}(p_k)}\subsetneqq\dotsb$.

Moreover, the isometric GIET can be suspended to a smooth foliation, without singularities, on a 2-manifold. The exceptional (respectively dense) orbits of the GIET give rise to exceptional (respectively dense) leaves of the foliation. Finite genus 2-manifolds cannot support orientable foliations with the considered dynamics.