Olga V. Pochinka | Searcharxiv

arXiv2019

On existence of a Morse energy function for topological flows with finite chain recurrent sets

We prove the existence of a continuous Morse energy function for an arbitrary topological flow with finite hyperbolic (in topological sense) chain recurrent set on a topological manifold of any dimension. This result is a partial solution of the Morse problem of existence of continuous Morse functions on any topological manifolds. Namely, we prove that a topological manifold admits a continuous Morse function if it admits a topological flow with finite hyperbolic chain recurrent set.

math.DS↗

arXiv2018

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

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 $\mathbb S^2\times\mathbb S^1$.

math.DS↗

arXiv2018

On Embedding of Multidimensional Morse-Smale Diffeomorphisms in Topological Flows

J.~Palis found necessary conditions for a Morse-Smale diffeomorphism on a closed $n$-dimensional manifold $M^n$ to embed into a topological flow and proved that these conditions are also sufficient for $n=2$. For the case $n=3$ a possibility of wild embedding of closures of separatrices of saddles is an additional obstacle for Morse-Smale cascades to embed into topological flows. In this paper we show that there are no such obstructions for Morse-Smale diffeomorphisms without heteroclinic intersection given on the sphere $S^n, \,n\geq 4$, and Palis's conditions again are sufficient for such diffeomorphisms.

math.DS↗