Morse–Smale diffemorphisms with non-wandering points with pairwise different indices on 3-manifolds

In this paper, we consider a class $G$ of orientation-preserving Morse-Smale diffeomorphisms $f$, which are defined on a closed 3-manifold $M^3$ and whose non-wandering set consists of four fixed points with pairwise different Morse indices. It follows from the results of the work of S. Smale and K. Meyer that all gradient-like flows with similar properties have a Morse energy function with four critical points of pairwise different Morse indices. This means that the supporting manifold $M^3$ for these flows admits a Heegaar decomposition of genus 1 and, therefore, it is homeomorphic to the lens space $L_{p, q}$. Despite the simple structure of the non-wandering set of diffeomorphisms in the class $G$, there are diffeomorphisms with wildly embedded separatrices. According to the results of V. Grines, F. Laudenbach, O. Pochinka, such diffeomorphisms do not have an energy function, and the question of the topology of their ambient manifold remains open. According to the results of V. Grines, E. Zhuzhoma and V. Medvedev, $M^3$ is homeomorphic to the lens space $L_{p, q}$ in the case of tame embedding of closures of one-dimensional separatrices of the diffeomorphism $f\in G$. Moreover, the wandering set of the diffeomorphism $f$ contains at least $p$ of non-compact heteroclinic curves. In this paper, a similar result is obtained for arbitrary diffeomorphisms of the class $G$. Diffeomorphisms from the class $G$ with wild embedding of one-dimensional separatrices are constructed on each lens space $L_{p, q}$. Such examples were previously known only on the 3-sphere. It is also established that the topological conjugacy of diffeomorphisms of class $G$ with a single non-compact heteroclinic curve is completely determined by the equivalence of Hopf knots, which are the projection of a one-dimensional saddle separatrix into the space of the orbits of the sink basin. Moreover, any Hopf knot $L$ is realized by such a diffeomorphism. In this sense, the result obtained is similar to classification of D. Pixton's diffeomorphisms obtained by Ch. Bonatti and V. Grines.

Lens space