Quasi-Energy Function for Morse–Smale 3-Diffeomorphisms with Fixed Points with Pairwise Different Indices

O. V. Pochinka ; E. A. Talanova

Matematičeskie zametki, Tome 115 (2024) no. 4, pp. 597-609

Résumé

The present paper is devoted to a lower bound for the number of critical points of the Lyapunov function for Morse–Smale 3-diffeomorphisms with fixed points with pairwise distinct indices. It is known that, in the presence of a single noncompact heteroclinic curve, the supporting manifold of the diffeomorphisms under consideration is a 3-sphere, and the class of topological conjugacy of such a diffeomorphism $f$ is completely determined by the equivalence class (there exist infinitely many of them) of the Hopf knot $L_{f}$, which is a knot in the generating class of the fundamental group of the manifold $\mathbb S^2\times \mathbb S^1$. Moreover, any Hopf knot is realized by some diffeomorphism of the class under consideration. It is known that the diffeomorphisms defined by the standard Hopf knot $L_0=\{s\}\times \mathbb S^1$ have an energy function, which is a Lyapunovfunction whose set of critical points coincides with the chain recurrent set. However, the set of critical points of any Lyapunov function of a diffeomorphism $f$ with a nonstandard Hopf knot is strictly greater than the chain recurrent set of the diffeomorphism. In the present paper, for the diffeomorphisms defined by generalized Mazur knots, a quasi-energy function has been constructed, which is a Lyapunov function with a minimum number of critical points.

Détail

Keywords: Morse–Smale diffeomorphism, Hopf knot.

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