Mathematical Sciences: Pressure and Local Entropy in Dynamical Systems

海顿教授将用局部熵函数研究动力系统。这一概念将应用于三个领域。首先，海顿教授将用它找到一个公式来确定双曲集的豪斯多夫维数。其次，他将用它来求高维动力系统的奇异谱。最后，一个更精细的工具将用于研究非吸引子的逃逸率。一个动力系统可以被认为是一个随时间演变的系统，它有一个明确的、定义良好的规律。例如，给定流体的初始状态，至少在理论上，人们可以使用Navier-Stokes方程来预测流体在以后任何时间的状态。然而，这种进化是如此复杂，甚至是混乱的，以至于人们必须寻找描述进化的一般方法，而不是真正解决方程。这是动力系统领域的领域，也是海顿教授研究背后的推动力。

Professor Haydn will study dynamical systems using a local entropy function. There are three areas where this concept will be applied. First, Professor Haydn will use it to find a formula which determines the Hausdorff dimension of a hyperbolic set. Second, he will use it to find the singularity spectrum of for higher dimensional dynamical systems. Finally, a somewhat finer tool will be used to investigate escape rates for non-attractors. A dynamical system can be thought of as a system which evolves in time with a definite well defined law. For example, given the initial state of a fluid, at least theoretically one can use the Navier-Stokes equations to predict the state of the fluid at any later time. However, this evolution is so complicated, even chaotic, that one must look for general ways of describing the evolution without actually solving the equations. This is the realm of the field of dynamical systems and the driving force behind the research of Professor Haydn.