Preprint A197/2003
Interval exchange transformations and foliations on infinite genus two-manifolds
Américo López | Gutierrez, Carlos | Hector, Gilbert
Keywords:
interval exchange transformations | foliations on surfaces | recurrence
For each one of the properties (a) - (c) below, there is an Isometric Generalized Interval Exchange Transformation (i.e. isometric giet) having such property: (a) nontrivial 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 finitely many orbits. (b) coexistence of dense orbits and exceptional orbits; (c) existence of a dense sequence of exceptional orbits $\{{\cal O}(p_k);k=1,2,\ldots\}$ such that $\overline{{\cal O}(p_1)}\subsetneqq\overline {{\cal O}(p_2)}\subsetneqq\ldots\subsetneqq\overline{{\cal O}(p_k)}\subsetneqq\ldots.$ Moreover, the isometric giet can be suspended to a smooth foliation, without singularities, on a 2-manifold. The exceptional (resp. dense) orbits of the giet give rise to a exceptional (resp. dense) leaves of the foliation. Finite genus 2-manifolds cannot support orientable foliations with the considered dynamics.