Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds
- 12 October 2012
- journal article
- research article
- Published by Pleiades Publishing Ltd in Proceedings of the Steklov Institute of Mathematics
- Vol. 278 (1), 27-40
- https://doi.org/10.1134/s0081543812060041
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 zeroand one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse-Smale diffeomorphism on a closed 3-manifold.Keywords
This publication has 11 references indexed in Scilit:
- The energy function for gradient-like diffeomorphisms on 3-manifoldsDoklady Mathematics, 2008
- New relations for Morse-Smale systems with trivially embedded one-dimensional separatricesSbornik: Mathematics, 2003
- Новые соотношения для систем Морса - Смейла с тривиально вложенными одномерными сепаратрисамиМатематический сборник, 2003
- Three-manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curvesTopology and its Applications, 2002
- Wild unstable manifoldsTopology, 1977
- On Morse-Smale dynamical systemsTopology, 1969
- On Irreducible 3-Manifolds Which are Sufficiently LargeAnnals of Mathematics, 1968
- Differentiable dynamical systemsBulletin of the American Mathematical Society, 1967
- Morse Theory. (AM-51)Published by Walter de Gruyter GmbH ,1963
- On Gradient Dynamical SystemsAnnals of Mathematics, 1961