On bifurcations that change the type of heteroclinic curves of a Morse-Smale 3-diffeomorphism.

In this paper, we consider the class $G$ of orientation-preserving Morse-Smale diffeomorphisms defined on a closed 3-manifold whose nonwandering set consists of exactly four points of pairwise distinct Morse indices. It is known that the two-dimensional saddle separatrices of any such diffeomorphism always intersect, and their intersection necessarily contains non-compact heteroclinic curves, but can also contain compact ones. The main result of the paper is the construction of a path in the space of diffeomorphisms connecting the diffeomorphism $ f \in G $ with the diffeomorphism $f'\in G$, which has no compact heteroclinic curves. This result is an important step in solving the open problem of describing the topology of 3-manifolds that admit gradient-like diffeomorphisms with wildly embedded saddle separatrices.

Keywords: Morse-Smale diffeomorphism, heteroclinic curves, invariant manifolds, arc in the space of diffeomorphisms.