2018/12/04 by В. З. Гринес, Evgeny V. Kruglov, Grines, Viacheslav Z. +5
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1812.01436
openalex publication_date 2018/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A topological classification of many classes of dynamical systems with\nregular dynamics in low dimensions is often reduced to combinatorial\ninvariants. In dimension 3 combinatorial invariants are proved to be\ninsufficient even for simplest Morse-Smale diffeomorphisms. The complete\ntopological invariant for the systems with a single saddle point on the\n3-sphere is the embedding of the homotopy non-trivial knot into the manifold\n mathbb S2\× mathbb S1. If a diffeomorphism has several saddle points\ntheir unstable separatrices form arcs frames in the basin of the sink and\ncircles frame in the orbits space. Thus, the type of embedding of the circles\nframe into mathbb S2\× mathbb S1 is a topological invariant for\ndiffeomorphisms of this kind and this type turns out to be the complete\ntopological invariant for some classes of Morse-Smale 3-diffeomorphisms.\nRecently it was shown that the problem of embedding of a 3-diffeomorphism into\na topological flow is interconnected with the properties of embedding of the\narcs frame into the 3-Euclidean space. In this paper we consider the criteria\nfor the tame embedding of an arcs frame into the 3-Euclidean space as well as\nfor the trivial embedding of circles frame into mathbb S2\× mathbb S1.\nWe apply this criteria to prove that frames of one-dimensional separatrices in\nbasins of sources of rough 3-diffeomorhisms with two-dimensional expanding\nattractor are tamely embedded and their spaces of orbits are trivial embeddings\nof circles frame into mathbb S2\× mathbb S1.\n