Generic robustness of spectral decompositions

We prove that given a compact n-dimensional boundaryless manifold M, n greater than or equal to 2, there exists a residual subset R of Diff(1) (M) such that if f is an element of R admits a spectral decomposition (i.e., the nonwandering set Q(f) admits a partition into a finite number of transitive compact sets), then this spectral decomposition is robust in a generic sense (tame behavior). This implies a C-generic trichotomy that generalizes some aspects of a two-dimensional theorem of Mane [Topology 17 (1978) 386-396]. Lastly, Palis [Asterisque 261 (2000) 335-347] has conjectured that densely in Diff(k) (M) diffeomorphisms either are hyperbolic or exhibit homoclinic bifurcations. We use the aforementioned results to prove this conjecture in a large open region of Diff(1) (M). (C) 2003 Editions scientifiques et medicales Elsevier SAS.