Geometric realization for substitution tilings
Geometric realization for substitution tilings
复制标题
替代平铺的几何实现
DOI：
10.1017/etds.2012.142
复制
发表时间：
2011
影响因子：
0.9
通讯作者：
J. Gambaudo
中科院分区：
文献类型：
--
作者：
M. Barge;J. Gambaudo
给定一个n维代换Φ，其线性展开式Λ是幺模双曲的，我们利用平铺空间ΩΦ的一维整数Čech上同调的元素构造了一个有限对一的半共轭G：ΩΦ→？D，称为几何实现，之间的替代诱导动力学和双曲环自同构的不变集。若Λ满足Pisot族条件，且广义返回向量模的秩等于Λ的广义度，则G是满射的，且与ΩΦ上的λ n-作用的极大等度连续因子上的映射重合.得到了Pisot代换猜想的高维推广：若Λ满足Pisot族条件，且ΩΦ的一维上同调秩等于Λ的广义次数，则ΩΦ上的π-作用具有纯离散谱.
Abstract Given an n-dimensional substitution Φ whose associated linear expansion Λ is unimodular and hyperbolic, we use elements of the one-dimensional integer Čech cohomology of the tiling space ΩΦ to construct a finite-to-one semi-conjugacy G:ΩΦ→?D, called a geometric realization, between the substitution induced dynamics and an invariant set of a hyperbolic toral automorphism. If Λ satisfies a Pisot family condition and the rank of the module of generalized return vectors equals the generalized degree of Λ, G is surjective and coincides with the map onto the maximal equicontinuous factor of the ℝn-action on ΩΦ. We are led to formulate a higher-dimensional generalization of the Pisot substitution conjecture: if Λ satisfies the Pisot family condition and the rank of the one-dimensional cohomology of ΩΦ equals the generalized degree of Λ, then the ℝn-action on ΩΦhas pure discrete spectrum.