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.