Abstract
The paper describes the topological structure of closed manifolds of dimension \(\ge4\) that admit Morse–Smale diffeomorphisms whose nonwandering sets contain arbitrarily many sink periodic points, arbitrarily many source periodic points, and two saddle periodic points. The underlying manifolds of Morse–Smale diffeomorphisms with fewer saddle periodic points are also described.
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V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, and O. V. Pochinka, “Classification of Morse–Smale systems and topological structure of the underlying manifolds,” Russian Math. Surveys 74 (1), 37–110 (2019).
S. Smale, “Morse inequalities for a dynamical system,” Bull. Amer. Math. Soc. 66, 43–49 (1960).
S. Smale, “Differentiable dynamical systems,” Bull. Amer. Math. Soc. 73, 747–817 (1967).
E. V. Zhuzhoma and V. S. Medvedev, “Global dynamics of Morse–Smale systems,” Proc. Steklov Inst. Math. 261, 112–135 (2008).
V. Z. Grines and O. V. Pochinka, Introduction to the Topological Classification of Diffeomorphisms on Two- and Three-Dimensional Manifolds (Regular and Chaotic Dynamics, Moscow–Izhevsk, 2011) [in Russian].
V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev, and O. V. Pochinka, “Global attractor and repeller of Morse–Smale diffeomorphisms,” Proc. Steklov Inst. Math. 271, 103–124 (2010).
V. Z. Grines, E. V. Zhuzhoma and V. S. Medvedev, “On Morse–Smale diffeomorphisms with four periodic points on closed orientable manifolds,” Math. Notes 74 (3), 352–366 (2003).
V. Medvedev and E. Zhuzhoma, “Morse–Smale systems with few non-wandering points,” Topology Appl. 160 (3), 498–507 (2013).
R. J. Daverman and G. A. Venema, Embeddings in Manifolds, in Grad. Stud. Math. (Amer. Math. Soc., Providence, RI, 2009), Vol. 106.
V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, and V. S. Medvedev, “On topology of manifolds admitting a gradient-like flow with a prescribed non-wandering set,” Siberian Adv. Math. 29 (2), 116–127 (2019).
A. V. Chernavskii, “Singular points of topological imbeddings of manifolds and the union of locally flat cells,” Soviet Math. Dokl. 7, 433–436 (1966).
J. C. Cantrell, “Almost locally flat embeddings of \(S^{n-1}\) in \(S^n\),” Bull. Amer. Math. Soc. 69, 716–718 (1963).
L. S. Pontryagin, Smooth Manifolds and Their Applications in Homotopy Theory (Nauka, Moscow, 1976) [in Russian].
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Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 361-369 https://doi.org/10.4213/mzm12718.
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Zhuzhoma, E.V., Medvedev, V.S. Underlying Manifolds of High-Dimensional Morse–Smale Diffeomorphisms with Two Saddle Periodic Points. Math Notes 109, 398–404 (2021). https://doi.org/10.1134/S000143462103007X
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DOI: https://doi.org/10.1134/S000143462103007X