On embedding of arcs and circles in 3-manifolds in an application to dynamics of rough 3-diffeomorhisms with two-dimensional expanding attractor

Abstract

A topological classification of many classes of dynamical systems with regular dynamics in low dimensions is often reduced to combinatorial invariants. In dimension 3 combinatorial invariants are proved to be insufficient even for simplest Morse-Smale diffeomorphisms. The complete topological invariant for the systems with a single saddle point on the 3-sphere is the embedding of the homotopy non-trivial knot into the manifold $\mathbb S^2\times\mathbb S^1$. If a diffeomorphism has several saddle points their unstable separatrices form arcs frames in the basin of the sink and circles frame in the orbits space. Thus, the type of embedding of the circles frame into $\mathbb S^2\times\mathbb S^1$ is a topological invariant for diffeomorphisms of this kind and this type turns out to be the complete topological invariant for some classes of Morse-Smale 3-diffeomorphisms. Recently it was shown that the problem of embedding of a 3-diffeomorphism into a topological flow is interconnected with the properties of embedding of the arcs frame into the 3-Euclidean space. In this paper we consider the criteria for the tame embedding of an arcs frame into the 3-Euclidean space as well as for the trivial embedding of circles frame into $\mathbb S^2\times\mathbb S^1$. We apply this criteria to prove that frames of one-dimensional separatrices in basins of sources of rough 3-diffeomorhisms with two-dimensional expanding attractor are tamely embedded and their spaces of orbits are trivial embeddings of circles frame into $\mathbb S^2\times\mathbb S^1$.