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On Realization of Topological Conjugacy Classes of Morse–Smale Cascades on the Sphere \(S^n\)

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Abstract

We consider the class \(G(S^n)\) of orientation-preserving Morse–Smale diffeomorphisms defined on the sphere \(S^n\) of dimension \(n\geq 4\) under the assumption that the invariant manifolds of different saddle periodic points are disjoint. For diffeomorphisms in this class, we describe an algorithm for constructing representatives of all topological conjugacy classes.

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Notes

  1. A \(k\)-dimensional manifold \(N^k\subset M^n\) without boundary is locally flat at a point \(x\in N^k\) if there exists a neighborhood \(U(x)\subset M^n\) of \(x\) and a homeomorphism \( \varphi:U(x)\to{\mathbb R}^n\) such that \( \varphi (N^k\cap U(x))={\mathbb R}^k\), where \({\mathbb R}^k=\{(x_1,\dots,x_n)\in{\mathbb R}^n\mid x_{k+1}=x_{k+2}=\dots=x_n=0\}\). If the local flatness condition is violated at some point \(x\in N^k\), then the manifold \(N^k\) is said to be wild, and the point \(x\) is called a wildness point.

  2. An automorphism of a graph is a one-to-one map on the vertex set that preserves the incidence relation.

  3. The Morse index of an equilibrium state \(q\) of a flow is the number equal to the dimension of the manifold \( W^{\textrm{u}} _q\).

  4. A diffeotopy is an isotopy \(H(x,t):X\times [0,1]\to Y\) such that for any \(t\in [0,1]\) the map \(H(x,t):X\times \{t\}\to Y\) is a diffeomorphism.

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Funding

This work is supported by the Russian Science Foundation under grant 17-11-01041, except for the proof of Lemma 1. The work on the proof of Lemma 1 is supported by the HSE Laboratory of Dynamical Systems and Applications and the Ministry of Science and Higher Education of the Russian Federation under grant 075-15-2019-1931.

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Authors and Affiliations

  1. National Research University Higher School of Economics, Bol’shaya Pecherskaya ul. 25/12, Nizhny Novgorod, 603155, Russia

    V. Z. Grines, E. Ya. Gurevich & V. S. Medvedev

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  1. V. Z. Grines
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  2. E. Ya. Gurevich
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  3. V. S. Medvedev
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Correspondence to V. Z. Grines.

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Grines, V.Z., Gurevich, E.Y. & Medvedev, V.S. On Realization of Topological Conjugacy Classes of Morse–Smale Cascades on the Sphere \(S^n\). Proc. Steklov Inst. Math. 310, 108–123 (2020). https://doi.org/10.1134/S0081543820050089

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