Analysis of orthogonality and of orbits in affine iterated function systems
Analysis of orthogonality and of orbits in affine iterated function systems
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DOI：
10.1007/s00209-007-0104-9
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发表时间：
2006-06
影响因子：
0.8
通讯作者：
D. Dutkay;P. Jorgensen
中科院分区：
文献类型：
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作者：
D. Dutkay;P. Jorgensen
我们引入了仿射迭代函数系统（AIFS）的对偶性，它自然是由传统调和分析背景下的群对偶性激发的。我们的仿射系统产生由收缩仿射映射迭代定义的分形。我们通过在两个方向上缩放来为此类系统构建对偶性：通过收缩迭代来形成小分形，通过涉及膨胀矩阵迭代的递归来形成大分形。我们所说的小分形是指一个支持哈钦森规范测度 μ 的紧凑吸引子，并且我们询问 μ 何时是谱测度，即何时希尔伯特空间具有指数的标准正交基 (ONB)。我们使用一对匹配的仿射系统进一步引入傅立叶对偶性。使用接下来的某些极端循环和膨胀矩阵的正幂，我们构建了大的分形，这些分形以空位傅里叶级数为模型，并用作 X 的光谱。我们的两个主要结果提供了简单的几何条件，使我们能够确定大的分形何时是 X 的谱。我们的结果又用 2 维和 3 维的具体谢尔宾斯基分形来说明。
We introduce a duality for affine iterated function systems (AIFS) which is naturally motivated by group duality in the context of traditional harmonic analysis. Our affine systems yield fractals defined by iteration of contractive affine mappings. We build a duality for such systems by scaling in two directions: fractals in the small by contractive iterations, and fractals in the large by recursion involving iteration of an expansive matrix. By a fractal in the small we mean a compact attractorXsupporting Hutchinson’s canonical measure μ, and we ask when μ is a spectral measure, i.e., when the Hilbert spacehas an orthonormal basis (ONB) of exponentials. We further introduce a Fourier duality using a matched pair of such affine systems. Using next certain extreme cycles, and positive powers of the expansive matrix we build fractals in the large which are modeled on lacunary Fourier series and which serve as spectra forX. Our two main results offer simple geometric conditions allowing us to decide when the fractal in the large is a spectrum forX. Our results in turn are illustrated with concrete Sierpinski like fractals in dimensions 2 and 3.