When shape matters: deformations of tiling spaces
When shape matters: deformations of tiling spaces
复制标题
当形状很重要时：瓷砖空间的变形
DOI：
--
复制
发表时间：
2003
影响因子：
0.9
通讯作者：
L. Sadun
中科院分区：
文献类型：
--
作者：
A. Clark;L. Sadun
我们研究平铺动力系统的动力学及其变形。如果两个镶嵌系统具有相同的组合，则镶嵌空间是同胚的，但它们的动力学性质可能不同。有一个自然的映射${mathcal I}$从参数空间的可能形状的瓷砖到H1的一个模型平铺空间，值在${mathbb R}^d$。两个在${mathcal I}$下具有相同图像的平铺空间是相互局部可导的（MLD）。当图像的差异是“渐近可忽略的”，那么平铺动力学是拓扑共轭的，但通常不是MLD。对于替代tilings，我们给出了一个简单的测试上同调类是渐近可忽略的，并表明，无穷小的形状变形的结果拓扑共轭动力学只有当图像的变化${mathcal I}$是渐近可忽略的。最后，我们给出了（变形的）置换镶嵌空间是拓扑弱混合的判据。
We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map ${mathcal I}$ from the parameter space of possible shapes of tiles to H1 of a model tiling space, with values in ${mathbb R}^d$. Two tiling spaces that have the same image under ${mathcal I}$ are mutually locally derivable (MLD). When the difference of the images is ‘asymptotically negligible’, then the tiling dynamics are topologically conjugate, but generally not MLD. For substitution tilings, we give a simple test for a cohomology class to be asymptotically negligible, and show that infinitesimal deformations of shape result in topologically conjugate dynamics only when the change in the image of ${mathcal I}$ is asymptotically negligible. Finally, we give criteria for a (deformed) substitution tiling space to be topologically weakly mixing.