Computational Tools for Dynamical Systems

在过去的几十年里，计算机在科学、工程和医学中的应用得到了极大的扩展。随着要解决的问题的复杂性和硬件解决这些问题的能力的增加，计算数学领域变得更加重要。当前计算工具的状态仍然落后于当前研究的状态。这个项目旨在通过解决公开的计算问题及其在计算机代数系统SAGE中的实现来缩小动力系统领域的差距。动力系统在整个科学领域有无数的应用。它们包括物理学中的时间逆转对称性，天气预报中的混沌系统，生物学和医学中的细胞和遗传过程，以及密码学和信息安全。随着动力系统研究人员的不断增加，对一套全面的计算工具的需求日益明显。这些工具将使研究人员能够快速有效地测试和完善理论，并计算复杂的例子。此外，学生可以使用这些工具来快速参与当前的研究问题。除了扩展动力系统的通用计算工具集，本项目还专门解决动力系统算法中的问题，以及由迭代函数引起的数论问题。特别令人感兴趣的是具有有限前向轨道的点和子簇，称为准周期。本项目旨在研究与数域上的有理准周期点和典范高度以及亚簇的迭代和典范高度相关的计算问题。这个项目的这两个方面都直接应用于后临界有限地图的工作。

Over the last several decades, the use of computers in the science, engineering, and medicine has expanded enormously. With the increasing complexity of problems to be solved and the increasing capacity of hardware to solve these problems, the field of computational mathematics became more essential. The state of current computational tools is still behind the state of current research. This project aims to narrow that gap in the area of dynamical systems by the resolution of open computational problems and their implementation in the computer algebra system: Sage. Dynamical systems have a myriad of applications throughout the sciences. They include time-reversing symmetries in physics, chaotic systems in weather prediction, cellular and genetic processes in biology and medicine, and cryptography and information security. With the increasing number of researchers in dynamical systems, there is a distinct need for a comprehensive set of computational tools. These tools will allow researchers to quickly and efficiently test and refine theories and to compute complicated examples. In addition, students can use these tools to quickly become involved in current research problems.In addition to expanding the general set of computational tools for dynamical systems, this project specifically addresses problems in the arithmetic of dynamical systems, and number theoretic questions arising from iterating functions. Of particular interest are points and subvarieties with finite forward orbits, called preperiodic. This project aims to study the computational problems associated with rational preperiodic points and canonical heights over number fields and iteration and canonical heights for subvarieties. Both aspects of this project have direct applications to working with post-critically finite maps.