Index in Dynamics: A Tool to Prove the Entropy Conjecture

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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.