ALGEBRAIC NUMBERS, FREE GROUP AUTOMORPHISMS AND SUBSTITUTIONS ON THE PLANE
ALGEBRAIC NUMBERS, FREE GROUP AUTOMORPHISMS AND SUBSTITUTIONS ON THE PLANE
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DOI：
10.1090/s0002-9947-2011-05188-3
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发表时间：
2011-09-01
影响因子：
1.3
通讯作者：
Ito, Shunji
中科院分区：
文献类型：
--
作者：
Arnoux, Pierre;Furukado, Maki;Ito, Shunji
最近有很多关于Pisot替换的几何表示和Pisot toral自同构的Markov划分的显式构造的工作。为了简单起见，我们考虑了一个简单的例子：自由群在4个生成元上的自同构，其关联矩阵有4个不同的复特征值，其中两个特征值的模大于1，另两个特征值的模小于1（非Pisot情形）.利用该生成器，我们构造了矩阵的收缩平面和扩张平面的替换多边形拼接。我们证明，这些替代平铺可以解除以一种独特的方式，以阶梯状的表面近似每个这些飞机。每一阶跃曲面的顶点都可以投影到另一个平面上的一个“原子曲面”上，原子曲面是一个具有分形边界的紧集.我们证明了这两个曲面都可以精化为具有分形边界的精确自相似曲面，并且可以以原子曲面为窗口，通过迭代或“切割投影”的方法得到.利用自相似曲面，人们可以建立一个与复的卡塔展开式相关联的数制;与这个数制相关联的卡塔展开式的自然扩展是通过自由群自同构的阿贝尔化获得的线性映射。这给出了这个双曲环自同构的显式Markov划分。分数维域可以用来定义一个伪平移群，该伪平移群给出Vershik（1994）意义上的横向动力学或Kamae（2005）意义上的计数系统。该构造可以扩展到更大类的自由群自同构，其中每一个都可以用于构建替换规则和动力系统。
There has been much recent work on the geometric representation of Pisot substitutions and the explicit construction of Markov partitions for Pisot toral automorphims. We give a construction that extends this to the general hyperbolic case.For the sake of simplicity, we consider a simple example of an automorphism of the free group on 4 generators whose associated matrix has 4 distinct complex eigenvalues, two of them of modulus larger than 1 and the other 2 of modulus smaller than 1 (non-Pisot case). Using this generator, we build substitution polygonal tilings of the contracting plane and the expanding plane of the matrix. We prove that these substitution tilings can be lifted in a unique way to stepped surfaces approximating each of these planes. The vertices of each of these stepped surfaces can be projected to an "atomic surface", a compact set with fractal boundary contained in the other plane.We prove that both tilings can be refined to exact self-similar tilings whose tiles have fractal boundaries and can be obtained by iteration or by a "cut and project" method by using the atomic surface as the window.Using the self-similar tiling, one can build a numeration system associated to a complex lambda-expansion; the natural extension of the lambda-expansion associated with this number system is the linear map obtained by abelianization of the free group automorphism. This gives an explicit Markov partition of this hyperbolic toral automorphism.The fractal domains can be used to define a pseudo-group of translations which gives transversal dynamics in the sense of Vershik (1994) or numeration systems in the sense of Kamae (2005).The construction can be extended to a larger class of free group automorphisms, each of which can be used to build substitution rules and dynamical systems.