Morse-Smale diffeomorphisms with non-wandering points of pairwise different Morse indices on 3-manifolds

Abstract

We consider the class $G$ of orientation-preserving Morse-Smale diffeomorphisms $f$ defined on a closed 3-manifold $M^3$, whose non-wandering set consists of four fixed points with pairwise different Morse indices. It follows from results due to Smale and Meyer that all gradient-like flows with similar properties have a Morse energy function with four critical points of pairwise distinct Morse indices. This means that the supporting manifold $M^3$ of such a flow admits a Heegaard decomposition of genus 1, so that it is diffeomorphic to a 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 results due to Grines, Laudenbach, and Pochinka, such diffeomorphisms have no energy function and the question of the topology of the supporting manifold is still open. According to results due to Grines, Zhuzhoma, and V. Medvedev, $M^3$ is homeomorphic to a lens space $L_{p,q}$ in the case of a tame embedding of the one-dimensional separatrices of the diffeomorphism $f\in G$. Moreover, the wandering set of $f$ contains at least $p$ non-compact heteroclinic curves. We obtain a similar result for arbitrary diffeomorphisms in the class $G$. On each lens space $L_{p,q}$ we construct diffeomorphisms from $G$ with wild embeddings of one-dimensional separatrices. Such examples were previously known only on the 3-sphere. We also show that the topological conjugacy of two diffeomorphisms in $G$ with a unique non-compact heteroclinic curve is fully determined by the equivalence of the Hopf knots that are the projections of one-dimensional saddle separatrices onto the orbit space of the sink basin. Moreover, each Hopf knot $L$ can be realized by such a diffeomorphism. In this sense the result obtained is similar to the classification of Pixton diffeomorphisms obtained by Bonatti and Grines. Bibliography: 65 titles.