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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Volume 278, Pages 34–48
(Mi tm3404)
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This article is cited in 14 scientific papers (total in 14 papers)
Dynamically ordered energy function for Morse–Smale diffeomorphisms on $3$-manifolds
V. Z. Grinesa, F. Laudenbachb, O. V. Pochinkaa a Lobachevsky State University of Nizhni Novgorod, Nizhni Novgorod, Russia
b Laboratoire de Mathématiques Jean Leray, UMR 6629 du CNRS, Faculté des Sciences et Techniques, Université de Nantes, Nantes, France
Abstract:
This paper deals with arbitrary Morse–Smale diffeomorphisms in dimension $3$ and extends ideas from the authors' previous studies where the gradient-like case was considered. We introduce a kind of Morse–Lyapunov function, called dynamically ordered, which fits well the dynamics of a diffeomorphism. The paper is devoted to finding conditions for the existence of such an energy function, that is, a function whose set of critical points coincides with the nonwandering set of the considered diffeomorphism. We show that necessary and sufficient conditions for the existence of a dynamically ordered energy function reduce to the type of the embedding of one-dimensional attractors and repellers, each of which is a union of zero- and one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse–Smale diffeomorphism on a closed $3$-manifold.
Received in March 2011
Citation:
V. Z. Grines, F. Laudenbach, O. V. Pochinka, “Dynamically ordered energy function for Morse–Smale diffeomorphisms on $3$-manifolds”, Differential equations and dynamical systems, Collected papers, Trudy Mat. Inst. Steklova, 278, MAIK Nauka/Interperiodica, Moscow, 2012, 34–48; Proc. Steklov Inst. Math., 278 (2012), 27–40
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https://www.mathnet.ru/eng/tm3404 https://www.mathnet.ru/eng/tm/v278/p34
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Abstract page: | 291 | Full-text PDF : | 55 | References: | 68 |
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