CADERNOS DE MATEM ´ ATICA 03, 77–97 May (2002) ARTIGO N ´ UMERO SMA#137 A structure theorem for foliations on non-compact two-manifolds Am´ ericoL´opez * Departamento de Matem´atica, Instituto de Ciˆ encias Matem´aticas e de Computa¸c˜ ao, Universidade de S˜ao Paulo - Campus de S˜ao Carlos, Caixa Postal 668, 13560-970 S˜ao Carlos SP, Brazil E-mail: amlopez@icmc.sc.usp.br We consider singular orientable foliations, which admit nontrivial recurrent leaves, on two-manifolds of finite or infinite genus. We give a structure theorem for this foliations. This one is similar to Gutierrez’s structure theorem [Gu1] for flows on compact surfaces. May, 2002 ICMC-USP 1. INTRODUCTION 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 smooth- ing 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 fo- liations 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 al- ready 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). * Partially supported by CNPq and FAPESP (Brazil), grant number 00/05144-0. 77 Publicado pelo ICMC-USP Sob a supervis˜ao CPq/ICMC