Mathematical Sciences: Measures, Dimension and Spectrum

点击翻译按钮获取中文摘要

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. ***