Email this article On the reduction of the topological classification of gradient-like flows problem to the classification of polar flows
- Authors: Saraev I.A.1
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Affiliations:
- Higher School of Economics
- Issue: Vol 25, No 2 (2023)
- Pages: 62-75
- Section: Mathematics
- Published: 25.05.2023
- URL: https://journal-vniispk.ru/2079-6900/article/view/358558
- DOI: https://doi.org/10.15507/2079-6900.25.202302.62-75
- ID: 358558
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Abstract
In this paper we consider a class G(Mn) of gradient-like flows on connected closed manifolds of dimension n ≥ 4 such that for any flow ft ∈ G(Mn) stable and unstable invariant manifolds of saddle equilibria do not intersect invariant manifolds of other saddle equilibria. It is known that the ambient manifold of any flow gft from the class G(Mn) can be splitted into connected summ of the sphere Sn, gft ≥ 0 copies of direct products Sn-1 x S1,and a simply connected manifold which is not homeomorphic to the sphere. The number gft is determined only by the number of nodal equilibria and the number of saddle equilibria such that one of their invariant manifolds has the dimension (n-1) (we call such equilibria trivial saddles). A simply connected manifold which is not homeomorphic to the sphere presents in the splitting if and only if the set of saddle equilibria contains points with unstable manifolds of dimension {2,...,n-2} (we call such equilibria non-trivial saddles). Moreover, the complete topological classification was obtained for flows from the class G(Mn) without non-trivial saddles. In this paper we prove that for any flow ft ∈ G(Mn) the carrier manifold can be splitted into a connected sum along pairwise disjoint smoothly embedded spheres (separating spheres) that do not contain equilibrium states of the flow ft and transversally intersect its trajectories. The restriction of the flow ft to the complements to these spheres uniquely (up to topological equivalence and numbering) defines a finite set of flows ft1,...,ftl defined on the components of a connected sum. Moreover, for any j ∈ {1,...,l}, the set of saddle equilibria of the flow ftj onsists either only of trivial saddles or only of of non-trivial ones and then the flow ftj is polar. We introduce the notion of consistent topological equivalence for flows j ∈ {1,...,l} and show that flows ft , f't∈ G(Mn) are topologically equivalent if and only if for each of these flows the set of separating spheres exists that defines consistently topologically equivalent flows on the components of the connected sum.
About the authors
Ilya A. Saraev
Higher School of Economics
Author for correspondence.
Email: isaraev@hse.ru
ORCID iD: 0000-0002-7608-2634
Student of the Faculty of Informatics, Mathematics and Computer Science, Trainee Researcher, Laboratory “Dynamical Systems and Applications”
Russian Federation, 25/12 B. Pecherskaya St., Nizhny Novgorod 603150, RussiaReferences
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