Links and dynamics

Abstract

Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the equivalence class of the Hopf knot, which is the orbit space of the unstable saddle separatrix in the manifold $\mathbb{S}^2\times \mathbb{S}^1$, is a complete invariant of the topological conjugacy of the system. In this paper we distinguish a class of three-dimensional Morse-Smale diffeomorphisms for which the complete invariant of topological conjugacy is the equivalence class of a link in $\mathbb{S}^2\times \mathbb{S}^1$. We proved that if $M$ is a link complement in $\mathbb{S}^3$ (in particular, is $\mathbb{S}^3$), or a handlebody $H_g$ of genus $g \geq 0$, or closed, connected, orientable 3-manifold, then the set of equivalence classes of tame links in $M$ is countable. As corollary we get that in $\mathbb{S}^2\times \mathbb{S}^1$ there exists a countable number of equivalence classes of tame links. It is proved that any essential link can be realized by a diffeomorphism of the class under consideration.