论文标题
自我捕捞2吸引者和瓷砖
Self-affine 2-attractors and tiles
论文作者
论文摘要
我们在$ \ mathbb {r}^d $中研究两位数的吸引子（2-aftractors），这是由两个收缩仿射映射与相同线性部分定义的自动紧凑型集合。由于近似理论中的许多应用，在多变量HAAR系统和其他小波底座的构建中，在离散的几何学和数量理论中，在文献中广泛研究了文献中的研究：Twindragons，Twindragons，Twindragons，两位数的瓷砖，2个reptiles等。我们在$ \ mathbb {r}^d $中获得了各向同性2-aftractors的完整分类，并证明它们都是同型但不是差异形态的。在一般的，非各向异性的情况下，证明了一个2吸引者是通过扩张矩阵的光谱来唯一定义的，直到仿射相似性。我们通过使用免费系数$ \ pm 2 $分析整数统一扩展多项式，估计$ \ mathbb {r}^d $中不同2 Attractors的数量。通过Mahler度量估算此类多项式的总数。我们介绍了几个无限的此类多项式。对于某些两个吸引者，可以找到他们的Hölder指数。我们的一些结果扩展到具有任意数字数字的吸引子。
We study two-digit attractors (2-attractors) in $\mathbb{R}^d$ which are self-affine compact sets defined by two contraction affine mappings with the same linear part. They are widely studied in the literature under various names: twindragons, two-digit tiles, 2-reptiles, etc., due to many applications in approximation theory, in the construction of multivariate Haar systems and other wavelet bases, in the discrete geometry, and in the number theory. We obtain a complete classification of isotropic 2-attractors in $\mathbb{R}^d$ and show that they are all homeomorphic but not diffeomorphic. In the general, non-isotropic, case it is proved that a 2-attractor is uniquely defined, up to an affine similarity, by the spectrum of the dilation matrix. We estimate the number of different 2-attractors in $\mathbb{R}^d$ by analysing integer unitary expanding polynomials with the free coefficient $\pm 2$. The total number of such polynomials is estimated by the Mahler measure. We present several infinite series of such polynomials. For some of the 2-attractors, their Hölder exponents are found. Some of our results are extended to attractors with an arbitrary number of digits.
