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 orbitsMoreover, 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.

Existence of minimal flows on nonorientable surfaces

“…'s in the literature, see for example Chacon transformations in [5] and the interesting way of modifying g.i.e.t. 's analysed by Gutierrez et al in [10], we have not found such examples when dealing with ∞-i.e.t. 's with flips.…”