Classification of Morse-Smale diffeomorphisms with the chain of saddles on 3-manifolds Ch. Bonatti V. Grines O. Pochinka ∗ Contents Introduction 2 0.1 Torus heteroclinic laminations of order n ...................... 3 0.2 Formulation of results ................................ 9 1 The proof of theorem 1 9 2 Cut and squeeze operations on the manifold N = S 2 × S 1 10 2.1 Cut and squeeze operation on the manifold N along nonpunctured torus .... 10 2.2 Cut and squeeze operation on the manifold N along torus heteroclinic lamination of order n ....................................... 12 3 Dynamics of a diffeomorphism f ∈ G n 12 3.1 Limit behavior of invariant manifolds of saddle points ............... 12 3.2 Admissible system of neighborhoods ......................... 13 3.3 Interrelation between dynamics of a diffeomorphism f ∈ G and the orbit space of the group F .................................... 14 4 The proof of theorem 2 14 5 The proof of theorem 3 15 5.1 The proof of the necessity part of theorem 3 .................... 15 5.2 Sketch of the proof for sufficient part of theorem 3 ................. 15 6 Sketch of the proof for theorem 4 20 6.1 Attachment of saddle points ............................. 21 6.2 Attachment of node points .............................. 22 References 22 ∗ Research partially supported by RFBR 05-01-00501 (Russia) and CNRS (France). 1