Renormalization of polygon exchange maps arising from corner percolation
Renormalization of polygon exchange maps arising from corner percolation
复制标题
由角渗透引起的多边形交换图的重整化
DOI：
--
复制
发表时间：
2011
期刊：
影响因子：
--
通讯作者：
W. Patrick Hooper
中科院分区：
文献类型：
--
作者：
W. Patrick Hooper
本文描述了由正方形$[0，{frac{1}{2}}]中的点（α，β）参数化的多边形交换变换族{<$α，β} imes[0，{frac{1}{2}}]$。当α和β是无理数时，<$α，β有任意大周期的周期轨道。我们表明，几乎所有的参数，多边形交换映射的属性，几乎每个点是周期性的。然而，有一组密集的无理参数，这是失败的。通过仔细选择参数，可以使非周期点的测度任意接近于全测度。这些结果是由重整化的概念，在更一般的设置。也就是说，我们考虑了角点逾渗模型产生的平铺的重整化。
We describe a family {Ψα,β} of polygon exchange transformations parameterized by points (α,β) in the square $[0, {frac{1}{2}}] imes[0, {frac{1}{2}}]$. Whenever α and β are irrational, Ψα,β has periodic orbits of arbitrarily large period. We show that for almost all parameters, the polygon exchange map has the property that almost every point is periodic. However, there is a dense set of irrational parameters for which this fails. By choosing parameters carefully, the measure of non-periodic points can be made arbitrarily close to full measure. These results are powered by a notion of renormalization which holds in a more general setting. Namely, we consider a renormalization of tilings arising from the Corner Percolation Model.