arXiv subjects

Vyacheslav Grines

Publications and source records attributed to Vyacheslav Grines.

2 recordsLinked to original sources

arXiv2023

On homeomorphisms of three-dimensional manifolds with pseudo-Anosov attractors and repellers

The present paper is devoted to a study of orientation-preserving homeomorphisms on three-dimensional manifolds with a non-wandering set consisting of a finite number of surface attractors and repellers. The main results of the paper relate to a class of homeomorphisms for which the restriction of the map to a connected component of the non-wandering set is topologically conjugate to an orientation-preserving pseudo-Anosov homeomorphism. The ambient $Ω$-conjugacy of a homeomorphism from the class with a locally direct product of a pseudo-Anosov homeomorphism and a rough transformation of the circle is proved. In addition, we prove that the centralizer of a pseudo-Anosov homeomorphisms consists of only pseudo-Anosov and periodic maps.

math.DS↗

arXiv2021

Morse index of saddle equilibria of gradient-like flows on connected sums of $\mathbb{S}^{n-1}\times \mathbb{S}^1$

Let $M$ be either $n$-sphere $\mathbb{S}^{n}$ or a connected sum of finitely many copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$, $n\geq4$. A flow $f^t$ on $M$ is called gradient-like whenever its non-wandering set consists of finitely many hyperbolic equilibria and their invariant manifolds intersects transversally. We prove that if invariant manifolds of distinct saddles of a gradient-like flow $f^t$ on $M$ do not intersect each other (in other words, $f^t$ has no heteroclinic intersections), then for each saddle of $f^t$ its Morse index (i.e. dimension of the unstable manifold) is either $1$ or $n-1$, so there are no saddles with Morse indices $i\in\{2,\ldots,n-2\}$.

math.DS↗