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oai:mountainscholar.org:10217/883572025-12-30T03:16:40Zcom_10217_100532com_10217_100000com_10217_100468com_10217_100303col_10217_100538col_10217_100469

Motta, Francis Charles, author Shipman, Patrick D., advisor Bradley, R. Mark, committee member Cavalieri, Renzo, committee member Dangelmayr, Gerhard, committee member 2007-01-03T05:56:29Z 2007-01-03T05:56:29Z 2012 Motta_colostate_0053N_11052.pdf Motta_colostate_0053N_11052_suppl.zip http://hdl.handle.net/10217/88357 https://doi.org/10.25675/3.018396 Zip file contains supplementary video, no sound. 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. born digital masters theses ZIP MPEG English eng Colorado State University. Libraries 2000-2019 Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. dynamical systems ion bombardment topology patterns persistent homology Patterns in dynamics Text Master of Science (M.S.) Masters Mathematics Colorado State University This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). open.access