论文标题
一维平铺空间上的旋转数字和旋转类
Rotation numbers and rotation classes on one-dimensional tiling spaces
论文作者
论文摘要
我们将圆形图的旋转理论扩展到瓷砖空间。具体而言，我们考虑一个具有有限局部复杂性和研究的一维平铺空间$ω$ 自动地图$ f $符合身份，其位移强烈的模式（SPE）。代替熟悉的旋转编号，我们定义了同时的$ [μ] $。我们证明了这一类的存在和独特性结果，发展了非理性的概念，并证明了Poncaré定理的类似物：如果$ [μ$是非理性的，那么$ f $是在$ω______________________________________$上的均匀翻译上的均匀翻译。在这种情况下，$ f $是$ω$本身的半偶联到均匀翻译，并且只有$ [μ] $才在于$ω$的第一个同胞组的某个子空间。
We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a 1-dimensional tiling space $Ω$ with finite local complexity and study self-maps $F$ that are homotopic to the identity and whose displacements are strongly pattern equivariant (sPE). In place of the familiar rotation number we define a cohomology class $[μ]$. We prove existence and uniqueness results for this class, develop a notion of irrationality, and prove an analogue of Poncaré's Theorem: If $[μ]$ is irrational, then $F$ is semi-conjugate to uniform translation on a space $Ω_μ$ of tilings that is homeomorphic to $Ω$. In such cases, $F$ is semi-conjugate to uniform translation on $Ω$ itself if and only if $[μ]$ lies in a certain subspace of the first cohomology group of $Ω$.
