论文标题
自相似的二维次要移位的三个特征
Three characterizations of a self-similar aperiodic 2-dimensional subshift
论文作者
论文摘要
本章的目的是说明斐波那契单词的概括为$ \ mathbb {z}^2 $上的二维配置的情况。 More precisely, we consider a particular subshift of $\mathcal{A}^{\mathbb{Z}^2}$ on the alphabet $\mathcal{A}=\{0,\dots,15\}$ for which we give three characterizations: as the subshift $\mathcal{X}_Φ$ generated by a 2-dimensional morphism $ \ Mathcal {a} $定义的$φ$;当Wang Shift $ω__\ Mathcal {Z} $由16 Wang Tiles的集合$ \ Mathcal {Z} $定义；作为符号动力系统$ \ MATHCAL {X} _ {\ MATHCAL {p} _ \ MATHCAL {Z}，R_ \ MATHCAL {Z}} $代表ORBITS，在某些$ \ Mathbb {Z}^2 $ -Action $ -Action $ r_ \ Mathcal proctation in of the Orbits上$ \ mathbb {t}^2 $，并由某些拓扑分区编码$ \ mathcal {p} _ \ Mathcal {z} $的$ \ Mathbb {t}^2 $ of 16多边形原子。我们证明了他们的公平$ω_\ MATHCAL {z} = \ MATHCAL {X} _或MATHCAL {X} _ {\ MATHCAL {\ MATHCAL {p} _ \ MATHCAL {Z}，R_ \ MATHCAL {Z}} $通过与自我simimility相吻合。 本章提供了通过研究Jeandel-Rao Wang Shift获得的四个不同文章的结果的横向阅读。它可以在一个地方聚集在Desubstitute Wang Shifts和Desubstitute代码上，以$ \ MATHBB {Z}^2 $ - actions的DeSubStitute代码，将其重点放在简单的二维自我相似的亚转移上。 Sagemath代码以查找标记图块并计算$ \ Mathbb {Z}^2 $ - 旋转的Rauzy诱导允许重现计算。本章包含许多练习，其解决方案最终提供了解决方案。
The goal of this chapter is to illustrate a generalization of the Fibonacci word to the case of 2-dimensional configurations on $\mathbb{Z}^2$. More precisely, we consider a particular subshift of $\mathcal{A}^{\mathbb{Z}^2}$ on the alphabet $\mathcal{A}=\{0,\dots,15\}$ for which we give three characterizations: as the subshift $\mathcal{X}_Φ$ generated by a 2-dimensional morphism $Φ$ defined on $\mathcal{A}$; as the Wang shift $Ω_\mathcal{Z}$ defined by a set $\mathcal{Z}$ of 16 Wang tiles; as the symbolic dynamical system $\mathcal{X}_{\mathcal{P}_\mathcal{Z},R_\mathcal{Z}}$ representing the orbits under some $\mathbb{Z}^2$-action $R_\mathcal{Z}$ defined by rotations on $\mathbb{T}^2$ and coded by some topological partition $\mathcal{P}_\mathcal{Z}$ of $\mathbb{T}^2$ into 16 polygonal atoms. We prove their equality $Ω_\mathcal{Z} =\mathcal{X}_Φ=\mathcal{X}_{\mathcal{P}_\mathcal{Z},R_\mathcal{Z}}$ by showing that they are self-similar with respect to the substitution $Φ$. This chapter provides a transversal reading of results divided into four different articles obtained through the study of the Jeandel-Rao Wang shift. It gathers in one place the methods introduced to desubstitute Wang shifts and to desubstitute codings of $\mathbb{Z}^2$-actions by focussing on a simple 2-dimensional self-similar subshift. SageMath code to find marker tiles and compute the Rauzy induction of $\mathbb{Z}^2$-rotations is provided allowing to reproduce the computations. The chapter contains many exercises whose solutions are provided at the end.
