We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a one-dimensional tiling space \(\Omega \) with finite local complexity and study self-maps F that are homotopic to the identity and whose displacements are strongly pattern equivariant. In place of the familiar rotation number, we define a cohomology class \([\mu ] \in {\check{H}}^1(\Omega , {\mathbb {R}})\). We prove existence and uniqueness results for this class, develop a notion of irrationality, and prove an analogue of Poincaré’s theorem: If \([\mu ]\) is irrational, then F is semi-conjugate to uniform translation on a space \(\Omega _\mu \) of tilings that is homeomorphic to \(\Omega \). In such cases, F is semi-conjugate to uniform translation on \(\Omega \) itself if and only if \([\mu ]\) lies in a certain subspace of \({\check{H}}^1(\Omega , {\mathbb {R}})\).

我们将圆图的旋转理论扩展到平铺空间。具体来说，我们考虑具有有限局部复杂性的一维平铺空间\（\ Omega \），并研究与恒等式同位并且其位移具有强烈模式等变的自映射F。代替熟悉的旋转数，我们在{\ check {H}} ^ 1（\ Omega，{\ mathbb {R}}）\）中定义一个同调类\（[\ mu] \ in。我们证明了此类的存在和唯一性结果，提出了非理性的概念，并证明了庞加莱定理的一个类似物：如果\（[\ mu] \）是非理性的，则F是半共轭的，从而对空间\（ \ Omega _ \ mu \）平铺到\（\ Omega \）。在这种情况下，˚F是半缀合物均匀翻译上\（\欧米茄\）本身当且仅当\（[\亩] \）在于具有一定子空间\（{\校验{H}} ^ 1（ \ Omega，{\ mathbb {R}}）\）。