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The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3manifold M 3 . These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of M 3 of a special type with respect to the considered diffeomorphism.Mathematics Subject Classification: 37B25, 37D15, 57M30.

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Topological classification of even the simplest Morse-Smale diffeomorphisms on 3manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies primarily in the possible "wild" behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may be not only a point as in the two-dimensional case, but also a curve, compact or non-compact. The

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