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On embedding of arcs and circles in 3-manifolds in an application to dynamics of rough 3-diffeomorhisms with two-dimensional expanding attractor | | Viacheslav Z. Grines
; Evgeny V. Kruglov
; Timur V. Medvedev
; Olga V. Pochinka
; | | Date: |
4 Dec 2018 | | 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
$mathbb S^2 imesmathbb S^1$. 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 $mathbb S^2 imesmathbb S^1$ 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 $mathbb S^2 imesmathbb S^1$.
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 $mathbb S^2 imesmathbb S^1$. | | Source: | arXiv, 1812.1436 | | Services: | Forum | Review | PDF | Favorites | |
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