Computational Tools for Dynamical Systems

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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.