Patterns in dynamics
Date
2012
Authors
Motta, Francis Charles, author
Shipman, Patrick D., advisor
Bradley, R. Mark, committee member
Cavalieri, Renzo, committee member
Dangelmayr, Gerhard, committee member
Journal Title
Journal ISSN
Volume Title
Abstract
In this paper we introduce and explore the idea of persistent homology (PH) and discuss several applications of this computational topology tool beyond its intended purpose. In particular we apply persistence to data generated by dynamical systems. The application of persistent homology to the circle map will lead us to rediscover the well-known result about the distribution of points in the orbit of this ergodic system called the Three Distance Theorem. We then apply PH to data extracted from several models of ion bombardment of a solid surface. This will present us with an opportunity to discuss new ways of interpreting PH data by introducing statistics on its output. Using these statistics we will begin to develop a technique to answer questions of interest to physicists about the degree of ordering present in the topography of a solid surface after ion bombardment. Finally we observe some inherent limitations in PH and, through simple examples, develop techniques to improve the technology. Specically we will implement algorithms to iteratively spread points on a real algebraic variety and demonstrate that the methodology works to improve the signals in the output of PH.
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Subject
dynamical systems
ion bombardment
topology
patterns
persistent homology