论文标题
难以区分的渐近对和多维结构
Indistinguishable asymptotic pairs and multidimensional Sturmian configurations
论文作者
论文摘要
如果对于每种有限模式，每种配置中关联的事件集合至有限支持的$ \ MATHBB {Z}^d $的有限支持的允许允许，则在整个$ \ mathbb {z}^d $ -Shift上的两个渐近配置是无法区分的。我们证明，满足“翻转条件”的不可区分的渐近对，其特征是它们在有限连接的支撑物上的模式复杂性。此外，我们证明，满足翻转条件的均匀反复发生的渐近渐近对，由codimension-one（内部空间的维度）切割和项目方案描述，这与多维Sturmian构型相对应。这两个结果一起提供了对Sturmian序列对$ \ Mathbb {z}^d $的概括，其因子复杂性$ n+1 $。当前工作提出了许多开放问题，并在引言中列出。
Two asymptotic configurations on a full $\mathbb{Z}^d$-shift are indistinguishable if for every finite pattern the associated sets of occurrences in each configuration coincide up to a finitely supported permutation of $\mathbb{Z}^d$. We prove that indistinguishable asymptotic pairs satisfying a "flip condition" are characterized by their pattern complexity on finite connected supports. Furthermore, we prove that uniformly recurrent indistinguishable asymptotic pairs satisfying the flip condition are described by codimension-one (dimension of the internal space) cut and project schemes, which symbolically correspond to multidimensional Sturmian configurations. Together the two results provide a generalization to $\mathbb{Z}^d$ of the characterization of Sturmian sequences by their factor complexity $n+1$. Many open questions are raised by the current work and are listed in the introduction.
