Index in Dynamics: A Tool to Prove the Entropy Conjecture

许多问题，如机械物体的运动，化学反应，人口增长，以及市场上价格的动态变化，都可以用动态系统来建模，动态系统由状态的集合和时间演化定律来定义。有一个与动力系统相关的经典数，称为熵，它是系统复杂性的定量度量。动力系统中一个重要的问题，几十年来一直没有解决，被称为舒布熵猜想。它旨在利用系统的渐近信息来理解系统熵的下界。由于这一猜想的性质，需要从拓扑学和动力学的新的多学科工具，以充分解决这个猜想。 该项目涉及开发动力系统中的工具，称为指标理论，以提高我们对不动点周围动力学的一般知识，并着眼于解决熵猜想。由于这一建议整合了数学的几个分支的想法，它将进一步鼓励学生和教师之间的互动和合作。本项目的具体研究目标是利用指标理论研究以下项目：1）等价解析流之间的熵非退化性; 2）解析映射周期点增长率与Lefschetz增长率的关系; 3）球面上具有充分正则性的映射周期点增长率与度的关系。 通过这三个项目，我们将展示如何使用指数来获得熵的非简并性，并展示指数理论在寻找周期轨道增长率的拓扑下界方面是多么强大。 由于这三个方案都具有很强的拓扑性质，因此它们都与熵猜想密切相关。 该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many problems such as the motion of mechanical objects, chemical reactions, population growth, and the dynamics of the prices on the market can be modeled by dynamical systems, which are defined by a collection of states and a time-evolution law. There is a classical number associated with a dynamical system, called its entropy, that is a quantitative measure of the complexity of the system. An important problem in dynamical systems that has been unsolved for many decades is called Shub's entropy conjecture. It aims at understanding the lower bound of the entropy of a system using asymptotic information of the system. Due to the nature of this conjecture new multi-disciplinary tools from topology and dynamics are needed to fully solve the conjecture. This project is concerned with developing a tool in dynamical systems, called index theory, in order to advance our general knowledge about the dynamics around fixed points, and with an eye to solving the entropy conjecture. Since this proposal integrates ideas from several branches of mathematics, it will further encourage interactions and collaborations among students and faculty. The specific research goal of this project is to study the following projects using index theory: 1) entropy non-degeneracy between equivalent analytic flows; 2) the relationship between the growth rate of periodic points and the growth rate of the Lefschetz for analytic maps; 3) the relationship between the growth rate of periodic points and the degree for maps with sufficient regularity on a sphere. Through these three projects, we will show how to use the index to get entropy non-degeneracy and present how powerful index theory is in finding a topological lower bound for the growth rate of periodic orbits. Since the three projects are all of a strong topological flavor, these projects are closely related to the entropy conjecture. As a package, they will provide potential approaches and tools for the entropy conjecture.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.