A Spectral Sequence for the K-theory of Tiling Spaces

Spaces of tilings, finite telescopic approximations and gap-labelling

For a large class of tilings, including the Penrose tiling in two dimension as well as the icosahedral ones in 3 dimension, the continuous hull of such a tiling inherits a minimal lamination structure with flat leaves and a transversal which is a Cantor set. In this case, we show that the continuous hull can be seen as the projective limit of a suitable sequence of branched, oriented and flat compact manifolds.The algebraic topological features related to this sequence reflect the dynamical properties of the action on the continuous hull. In particular the set of positive invariant measures of…

关联非周期镶嵌的衍射与谱数据：走向 Bloch 定理

Relating Diffraction and Spectral Data of Aperiodic Tilings: Towards a Bloch theorem

本文揭示了非周期镶嵌在所有维度下由 Čech cohomology 描述的衍射图案与由 K-theory 定义的 Hamiltonians 谱性质之间的关系，证明二者在维度 $\leq 3$ 时等价，从而将 Bloch Theorem 推广至非周期镶嵌。

基于 $AF$-代数的二维替换铺嵌空间的 $K$-理论

$K$-theory of two-dimensional substitution tiling spaces from $AF$-algebras

本文证明在合理假设下，二维替换铺嵌空间不稳定广群 $C^\ast$-代数的 $K$-理论可由替换规则及其 $1$-骨架类似物的 $AF$-代数 $K$-理论显式重构，通过相对 $K$-理论与切除转化为纯拓扑正合序列，并给出 Sage 计算示例。