Symbolic Dynamics, Smooth Dynamics, and Applications

该项目有几个相关的组成部分：1)我们计划继续把谢林分离模型作为一个动力系统来研究。该模型最早出现在经济学领域，与许多晶格模型有关，如晶格气体，但由于格点耦合的固有非局域性质，该模型更加困难；2)我们计划研究符号和双曲动力系统周期点不变量的“刚性”。对于Holder连续函数f，拓扑不变量包括无标记周期轨道谱、Beta函数P(-S f)和Zeta函数。这些不变量是动力学和统计物理中的基本研究对象，但关于它们俘获的函数的信息是微妙的，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，Lyapunov指数、局部熵和Birkhoff平均)以及相应相空间组成的精细结构。这个模型最早是由著名经济学家托马斯·谢林提出的，它与统计物理中的许多格子模型有关，如格子气体，但由于格点耦合的固有非局域性质，该模型更加困难；2)压力是统计物理中的一个基本研究对象，但即使在高度理想化的系统中，关于它捕获的系统的信息也是微妙的，而且理解得很少。我们计划研究某些系统是否完全由它们的压强来确定。这些问题与Kac巧妙地总结为“你能听到鼓的形状吗？”的问题有着惊人的相似之处；(3)在遍历系统中，函数沿几乎每条轨道的时间平均等于空间平均。只有极少数情况下，几乎所有的轨道都可以被所有的轨道取代。我们计划研究时间平均与空间平均不重合的例外集的精细结构和维度

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