Symbolic Dynamics, Smooth Dynamics, and Applications

本研究计画包含以下几个相关部分：1）我们计划继续研究谢林分凝模型作为一个动力系统。这个模型，首先在经济学中出现，与统计物理学中的一些格子模型（如格子气）有关，但由于位置耦合的固有非局部性质而更加困难; 2）我们计划研究符号和双曲动力系统的周期点不变量的“刚性”。拓扑不变量包括，对于一个保持器连续函数f，未标记的周期轨道谱，β函数P（-s f），和zeta函数。这些不变量是动力学和物理学的基本研究对象，但它们所捕获的关于函数f的信息是微妙的，人们对它们的理解很少; 3）我们计划继续研究遍历理论中基本量的值的分布（例如，Lyapunovexponents，局部熵，和Birkhoff平均值）以及相应相空间组成的精细结构。拟议项目有几个相关组成部分：1）我们计划继续把谢林分凝模型作为一个动力学系统来研究。这个模型是由著名的经济学家托马斯谢林首先提出的，它与统计物理学中的一些格子模型（如格子气）有关，但由于位置耦合的内在非局部性质，它更加困难; 2）压力是统计物理学研究的一个基本对象，但即使在高度理想化的系统中，它所捕获的关于压力的信息也是微妙的，人们对它的理解也很有限。我们计划研究某些系统是否完全通过它们的压力来识别。这些问题与卡茨巧妙地总结为“你能听到鼓的形状吗？（3）Forergodic系统，函数沿沿着几乎每一轨道的时间平均等于空间平均。只有在极少数情况下，几乎每个轨道都可以被每个轨道所取代。我们打算研究时间平均与空间平均不一致的异常集的精细结构和维数

The proposed project has several related components: 1) Weplan to continue our study of the Schelling segregationmodel as a dynamical system. This model, which first arosein economics, is related to a number of lattice models instatistical physics like the lattice gas, but more difficultdue to the inherent non-local nature of site coupling; 2) Weplan to study the "rigidity" of periodic point invariantsfor symbolic and hyperbolic dynamical systems. Thesetopological invariants include, for a Holder continuousfunction f, the unmarked periodic orbit spectrum, the betafunction P(-s f), and the zeta function. These invariantsare fundamental objects of study in dynamics and statisticalphysics, but the information about the function f theycapture is subtle and poorly understood; 3) We plan tocontinue our investigation into the distribution of valuesof fundamental quantities in ergodic theory (e.g. Lyapunovexponents, local entropy, and Birkhoff averages) and thefine structure of the corresponding phase spacedecomposition.The proposed project has several related components: 1) Weplan to continue our study of the Schelling segregationmodel as a dynamical system. This model, which was firstproposed by the eminent economist Thomas Schelling, isrelated to a number of lattice models in statistical physicslike the lattice gas, but more difficult due to the inherentnon-local nature of site coupling; 2) Pressure is afundamental object of study in statistical physics, but evenin highly idealized systems, the information about thesystem it captures is subtle and poorly understood. We planto study whether certain systems are completely identifiedby their pressure. These problems have striking similaritiesto fascinating questions which Kac adroitly summarized withthe question "Can you hear the shape of a drum?"; (3) Forergodic systems, the time average of a function along almostevery orbit equals the spatial average. Only very rarely canalmost every orbit be replaced by every orbit. We plan tostudy the fine structure and dimension of the exceptionalset whose time average does not coincide with the spatialaverage