Index in Dynamics: A Tool to Prove the Entropy Conjecture

动力学索引：证明熵猜想的工具

基本信息

批准号：
2000167
负责人：
Yun Yang
金额：
$ 14.04万
依托单位：
Virginia Polytechnic Institute and State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2000167&HistoricalAwards=false
关键词：
Index Dynamics Tool Prove Entropy

项目摘要

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.

许多问题，例如机械物体的运动、化学反应、人口增长和市场价格动态，都可以通过动态系统来建模，动态系统由状态集合和时间演化定律定义。有一个与动力系统相关的经典数字，称为熵，它是系统复杂性的定量度量。动力系统中几十年来一直未解决的一个重要问题称为舒布熵猜想。它的目的是利用系统的渐近信息来理解系统熵的下界。由于该猜想的性质，需要拓扑和动力学等新的多学科工具来完全解决该猜想。该项目致力于开发动力系统中的一种工具，称为指数理论，以增进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。由于该提案整合了数学多个分支的思想，因此将进一步鼓励学生和教师之间的互动与合作。本项目的具体研究目标是利用指数理论研究以下项目：1）等价解析流之间的熵非简并性； 2）解析图的周期点增长率与Lefschetz增长率之间的关系； 3) 球面上具有足够规律性的图，周期点的增长率与度之间的关系。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的强大作用。由于这三个项目都具有浓厚的拓扑色彩，因此这些项目都与熵猜想密切相关。作为一个整体，它们将为熵猜想提供潜在的方法和工具。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。