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首页> 外文期刊>Annales Henri Poincare >Tiling Deformations, Cohomology, and Orbit Equivalence of Tiling Spaces
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Tiling Deformations, Cohomology, and Orbit Equivalence of Tiling Spaces

机译：平铺空间的变形，协调和轨道等效

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摘要

We study homeomorphisms of minimal and uniquely ergodic tiling spaces with finite local complexity (FLC), of which suspensions of (minimal and uniquely ergodic) d -dimensional subshifts are an example, and orbit equivalence of tiling spaces with (possibly) infinite local complexity (ILC). In the FLC case, we construct a cohomological invariant of homeomorphisms and show that all homeomorphisms are a combination of tiling deformations, maps homotopic to the identity (known as quasi-translations), and local equivalences (MLD). In the ILC case, we construct a cohomological invariant in the so-called weak cohomology and show that all orbit equivalences are combinations of tiling deformations, quasi-translations, and topological conjugacies. These generalize results of Parry and Sullivan to higher dimensions. We also show that homeomorphisms (FLC) or orbit equivalences (ILC) are completely parametrized by the appropriate cohomological invariants. Finally, we show that, under suitable cohomological conditions, continuous maps between tiling spaces are homotopic to compositions of tiling deformations and either local derivations (FLC) or factor maps (ILC).
机译：我们研究了具有有限局部复杂性（FLC）的最小且友好的遍历平铺空间的同源形态，其中悬浮液（最小且友好）的Dimension Subswarfts是一个例子，瓦片空间与（可能）无限的局部复杂度的轨道等效物（ ILC）。在FLC案例中，我们构建了同源术的同学不变，并表明所有同源性都是平铺变形的组合，将同型同型的同型（称为准翻译）和局部等效性（MLD）。在ILC案例中，我们在所谓的弱协调学中构建一个协调不变，并表明所有轨道等效性都是平铺变形，准翻译和拓扑共轭的组合。这些普遍性的帕里和沙利文的结果较高。我们还表明同源性（FLC）或轨道等效命令（ILC）是完全参加适当的协调不变的参数化。最后，我们表明，在合适的协调条件下，平铺空间之间的连续映射是平铺变形和局部衍生（FLC）或因子图（ILC）的组成的同型映射。

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