... Annals of Mathematics, 158 (2003), 355–418 A C 1-generic dichotomy for diffeomorphisms: Weak forms of hyperbolicity or infinitely many sinks or sources By C Bonatti, L J D´ ıaz, and E R Pujals* ... Abedenur, Marie-Claude Arnaud, Jairo Bochi, Michel Herman and Gioia Vago, for their careful reading of this paper, and to the students of the Dynamical Systems Seminar of IMPA for many comments ... surfaces are either Anosov or have Lyapunov exponents equal to zero for almost every orbit (see ne also [Ma4] for an outline of a possible proof).5 Ma˜´ also announced a version of his theorem...