Mathematical Sciences: Measures, Dimension and Spectrum

小行星9500744 该方案涉及到Bernoulli卷积测度、自相似分形集和平铺动力系统的研究。在过去的15年里，无限伯努利卷积在光滑动力学和维数理论中变得重要。利用最近的进展，了解这些措施，该项目将启动新的方向的攻击几个相互关联的问题，其中一个问题的Palis和Takens形态的算术差异的康托集。Tilings被用来作为模型的准晶和知识，他们的光谱是很重要的应用。该项目涉及开发一个框架，研究一般层次平铺，其中将包括一个自相似平铺的概念。旧的例子和建设的新的例子的频谱特性的调查将使用遍历理论和调和分析。最后，将调查的一个著名的例子，二维移动的有限型-“黄金分割”的转变。我们的目标是计算这种移动的熵。 遍历理论一般涉及理解系统的平均行为，其动力学过于复杂或混乱，以至于无法在微观细节中遵循。倾斜和重复卷积可以被看作是动力系统。这样，遍历理论就与几何学和分析学相联系了。这个项目涉及遍历理论问题沿着与这些领域的接口。 ***

9500744 Solomyak This proposal involves the investigation of Bernoulli convolution measures, self-similar fractal sets and tiling dynamical systems. In the last 15 years, infinite Bernoulli convolutions have become important in smooth dynamics and dimension theory. Using recent progress in understanding these measures, the project will initiate new directions of attack on several interrelated problems, among them a problem of Palis and Takens on the morphology of arithmetic differences of Cantor sets. Tilings are used as models of quasicrystals and a knowledge of their spectrum is important for applications. The project involves developing a framework for the study of general hierarchical tilings which would include the notions of a self-similar tiling. An investigation of spectral properties of old examples and the construction of new examples will be made using ergodic theory and harmonic analysis. Finally an investigation will be made of a well-known example of two-dimensional shift of finite type - the "golden mean" shift. The goal is to compute the entropy of this shift. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Tilings and repeated convolutions can be viewed as dynamical systems. In this way, ergodic theory makes contact with geometry and analysis. This project involves ergodic theory problems along the interface with these areas. ***