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、人口增长和市场价格的动态，都可以用动态系统来建模，这些系统由一组状态和一个时间演化定律定义。有一个与动力系统相关的经典数字，称为它的熵，它是对系统复杂性的定量测量。动力学系统中一个几十年来一直没有解决的重要问题被称为舒布熵猜想。它的目的是利用系统的渐近信息来理解系统的熵的下界。由于这一猜想的性质，需要新的拓扑学和动力学多学科工具来完全解决这一猜想。这个项目致力于开发动力系统中的一个工具，称为指数理论，以提高我们对不动点附近的动力学的一般知识，并着眼于解决熵猜想。由于这项提议融合了数学的几个分支的想法，它将进一步鼓励学生和教职员工之间的互动和合作。本课题的具体研究目标是利用指数理论研究如下问题：1)等价解析流之间的非退化熵；2)周期点的增长率与解析映射的Lefschetz增长率之间的关系；3)周期点的增长率与球面上具有充分正则性的映射的度之间的关系。通过这三个项目，我们将展示如何利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。由于这三个项目都具有很强的拓扑性，所以这些项目都与熵猜想密切相关。作为一个整体，他们将为熵猜想提供潜在的方法和工具。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.