论文标题
图形dan les表面和厄尔贡拓扑
Graphes dans les surfaces et ergodicité topologique
论文作者
论文摘要
用有限连接的图$ g $制造动态系统的最简单方法是使其具有极化，也就是说，对于每个顶点，将入围边缘的边缘循环排序。相位空间$ \ MATHCAL {p}（g）$由所有对$（v，e）$组成，其中$ v $是顶点，$ e $是$ v $的边缘事件。这样的初始条件给出了位置和动力。数据$（v，e）$当然等于赋予有方向$ e _ {\ Mathcal O} $的边缘。随着极化，每个初始数据都会导致通过在每个顶点向左转，或者如果没有其他边缘，则可以反弹。如果左路的所有边缘都在$ g $的所有边缘，不一定在两个方向上都可以完成。像往常一样，我们将顶点的价值定义为偶然的边缘数，并将图的价值定义为其顶点的价值的平均值。在本文中，我们证明，如果嵌入在封闭的属属$ g $的封闭式表面的图中，则可以完全向左步行，那么它的价最大为$ 1 + \ sqrt {6g + 1} $。我们证明，对于无限多个属$ g $，这一结果是敏锐的，并且它在$ g \ to + \ to + \ infty $上是渐近的最佳选择。这会导致表面上图在表面上的嵌入性的障碍，该方式可以完全允许完全向左步行。由于检查偏光图是否在$ 4N $上进行了全面的左步行，其中$ n $是边缘的基础性，因此这种障碍在可计算性方面特别有效。这个问题起源于有趣的后果，因为我们在这里称之为保守系统的奇迹性，尤其是二维的汉密尔顿系统$ h $，其中完整的左侧步行的存在与拓扑上的拓扑轨道相对应，即$ h $的拓扑范围，即$ h $访问表面的所有拓扑。
The simplest way to make a dynamical system out of a finite connected graph $G$ is to give it a polarization, that is to say a cyclic ordering of the edges incident to a vertex, for each vertex. The phase space $\mathcal{P}(G)$ then consists of all pairs $(v,e)$ where $v$ is a vertex and $e$ is an edge incident to $v$. Such an initial condition gives a position and a momentum. The data $(v,e)$ is of course equivalent to an edge endowed with an orientation $e_{\mathcal O}$. With the polarization, each initial data leads to a leftward walk defined by turning left at each vertex, or making a rebound if there is no other edge. A leftward walk is called complete if it goes through all edges of $G$, not necessarily in both directions. As usual, we define the valence of a vertex as the number of edges incident to it, and we define the valence of a graph as the average of the valences of its vertices. In this article, we prove that if a graph which is embedded in a closed oriented surface of genus $g$ admits a complete leftward walk, then its valence is at most $1 + \sqrt{6g+1}$. We prove furthermore that this result is sharp for infinitely many genera $g$, and that it is asymptotically optimal as $g \to + \infty$. This leads to obstructions for the embeddability of graphs on a surface in a way which admits a complete leftward walk. Since checking that a polarized graph admits a complete leftward walk or not is done in time $4N$, where $N$ is the cardinality of the edges, this obstruction is particularly efficient in terms of computability. This problem has its origins in interesting consequences for what we will call here the topological ergodicity of conservative systems, especially Hamiltonian systems $H$ in two dimensions where the existence of a complete leftward walk corresponds to a topologically ergodic orbit of the system, i.e. an orbit of $H$ visiting all the topology of the surface.
