Malyshev D, Morozov A, Pochinka O
Faculty of Informatics, Mathematics, and Computer Science, National Research University Higher School of Economics, Nizhny Novgorod 603155, Russian Federation.
Chaos. 2021 Feb;31(2):023119. doi: 10.1063/5.0029352.
In this paper, we consider a class of orientation-preserving Morse-Smale diffeomorphisms defined on an orientable surface. The papers by Bezdenezhnykh and Grines showed that such diffeomorphisms have a finite number of heteroclinic orbits. In addition, the classification problem for such diffeomorphisms is reduced to the problem of distinguishing orientable graphs with substitutions describing the geometry of a heteroclinic intersection. However, such graphs generally do not admit polynomial discriminating algorithms. This article proposes a new approach to the classification of these cascades. For this, each diffeomorphism under consideration is associated with a graph that allows the construction of an effective algorithm for determining whether graphs are isomorphic. We also identified a class of admissible graphs, each isomorphism class of which can be realized by a diffeomorphism of a surface with an orientable heteroclinic. The results obtained are directly related to the realization problem of homotopy classes of homeomorphisms on closed orientable surfaces. In particular, they give an approach to constructing a representative in each homotopy class of homeomorphisms of algebraically finite type according to the Nielsen classification, which is an open problem today.
在本文中，我们考虑一类定义在可定向曲面上的保定向莫尔斯 - 斯梅尔微分同胚。Bezdenezhnykh和Grines的论文表明，此类微分同胚具有有限数量的异宿轨道。此外，此类微分同胚的分类问题可归结为区分带有描述异宿相交几何结构的替换的可定向图的问题。然而，此类图通常不允许多项式判别算法。本文提出了一种对这些级联进行分类的新方法。为此，将所考虑的每个微分同胚与一个图相关联，该图允许构建一种用于确定图是否同构的有效算法。我们还确定了一类可允许图，其每个同构类都可以由具有可定向异宿的曲面的微分同胚来实现。所获得的结果与闭可定向曲面上同胚的同伦类的实现问题直接相关。特别地，它们给出了一种根据尼尔森分类在代数有限型同胚的每个同伦类中构造一个代表的方法，这在当今是一个未解决的问题。
