A structure theorem for foliations on non-compact 2-manifolds

Abstract

In this paper we study the structure of non-trivial recurrence leaves. The obtained results have been shown to be very useful in the smoothing of the continuous singular foliations on arbitrary genus two-manifolds [Lo1]. From the famous example of Denjoy [De], it is clear that it is very important to understand the dynamical structure of non-trivial recurrence. For compact two-manifolds, the study of non-trivial recurrence together with the smoothing problems for orientable singular foliations was carried out by C. Gutierrez [Gu1]. On the other hand, recent work shows that infinite genus surfaces can support orientable foliations whose recurrent leaves have dynamics which can not appear on compact surfaces (see [Gu-He-Lo]). Besides, the existence of minimal foliations on those surfaces was already proven by J. Beniere (see [Be]). Our main theorem generalize, to orientable singular foliations on two-manifolds of infinite genus, C. Gutierrez’s structure theorem [Gu1]. To state the theorem, we need some definitions. Let T : R/Z→ R/Z be a map whose domain of definition (Dom(T )) and image (Im(T )) are open and dense subsets of R/Z. We say that T is a generalized interval exchange transformation(or shortly a GIET) if T takes homeomorphically each connected component of its domain of definition onto a connected component of its image. A GIET is said to be affine (resp. isometric) if it restricted to every connected component of its domain of definition, is affine (resp. isometric). If T is an isometric GIET such that R/Z Dom(T ) is at most finite, then we shall say that T is a standard interval exchange transformation (standard IET).