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Taurida Journal of Computer Science Theory and Mathematics, 2021, Issue 1, Pages 101–114
(Mi tvim112)
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This article is cited in 1 scientific paper (total in 1 paper)
On bifurcations that change the type of heteroclinic curves of a Morse-Smale 3-diffeomorphism
V. I. Shmukler, O. V. Pochinkaa a National Research University – Higher School of Economics in Nizhny Novgorod
Abstract:
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∈G with the diffeomorphism f′∈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 M3, the non-wandering set of which consists of exactly four points ω,σ1,σ2,α 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∈G the set Hf=Wsσ1∩Wuσ2 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∈G, the bearing manifold M3 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 M3 admit a diffeomorphism f∈G. Then the same manifold admits a diffeomorphism f′∈G, a wandering set that does not contain compact heteroclinic curves.
Keywords:
Morse-Smale diffeomorphism, heteroclinic curve, unstable manifold, stable manifold, orientation-preserving diffeomorphism manifold, topological flow, regular dynamics, hyperbolic set, chain recurrent set.
Citation:
V. I. Shmukler, O. V. Pochinka, “On bifurcations that change the type of heteroclinic curves of a Morse-Smale 3-diffeomorphism”, Taurida Journal of Computer Science Theory and Mathematics, 2021, no. 1, 101–114
Linking options:
https://www.mathnet.ru/eng/tvim112 https://www.mathnet.ru/eng/tvim/y2021/i1/p101
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