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

V. I. Shmukler ; O. V. Pochinka

Taurida Journal of Computer Science Theory and Mathematics, no. 1 (2021), pp. 101-114

Résumé

In this paper, we consider the class $G$ of orientation-preserving Morse-Smale diffeomorphisms defined on a closed $3$-manifold whose non-wandering 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 may also contain compact ones. The main result of this work is the construction of a path in the space of diffeomorphisms connecting the diffeomorphism $ f \in G $ with the diffeomorphism $f' \in G$, which does not have compact heteroclinic curves. This result is an important step in solving the open problem of describing the topology of $3$-manifolds admitting gradient-like diffeomorphisms with wildly embedded saddle separatrices. Consider the class $G$ of orientation-preserving Morse-Smale diffeomorphisms $f$ defined on the closed manifold $ M^3$, the non-wandering set of which consists of exactly four points $ \omega, \sigma_{1}, \sigma_{2}, \alpha $ with positive types of orientation and with Morse indices (dimensions of unstable manifolds) $ 0,1,2,3 $, respectively. Despite the simple structure of the non-wandering set, the class under consideration contains diffeomorphisms with wildly embedded saddle separatrices [2] (see Fig. 1). It was proved in [3] that for any diffeomorphism $ f \in G $ the set $ H_f = W^{s}_{\sigma_{1}} \cap {W^{u} _ { \sigma_{2}}} $ is not empty and contains at least one non-compact heteroclinic curve. According to [3], in the case of a manual embedding of the closures of one-dimensional separatrices of the diffeomorphism $f \in G$, the bearing manifold $ M^3 $ admits a Heegaard decomposition of genus $1$ and, therefore, is a lens space (see, for example, [7]). In the case of a wild embedding, the description of the topology of the supporting manifold is an open problem formulated in [3]. In the present paper, an important step has been taken in solving this problem; namely, the following fact is proved. Theorem. Let the manifold $ M^3 $ admit a diffeomorphism $ {f} \in {G} $. Then the same manifold admits a diffeomorphism $ {f}'\in {G} $, a wandering set that does not contain compact heteroclinic curves.

Détail

Keywords: Morse-Smale diffeomorphism, heteroclinic curve, unstable manifold, stable manifold, orientation-preserving diffeomorphism manifold, topological flow, regular dynamics, hyperbolic set, chain recurrent set.