{"id":49,"date":"2020-02-25T08:30:26","date_gmt":"2020-02-25T08:30:26","guid":{"rendered":"http:\/\/sites.msudenver.edu\/niemeye1\/?page_id=49"},"modified":"2020-02-25T09:11:52","modified_gmt":"2020-02-25T09:11:52","slug":"fractal-billiards","status":"publish","type":"page","link":"https:\/\/sites.msudenver.edu\/niemeye1\/research-activities\/fractal-billiards\/","title":{"rendered":"Fractal billiards, flat surfaces, IETs"},"content":{"rendered":"<p style=\"padding-left: 40px\"><span style=\"font-size: 14pt\"><a href=\"http:\/\/ccjohnson.org\/\">C. Johnson<\/a> and I are currently investigating the flow on a fractal flat surface. Related to this is another joint project concerning the existence of a saddle connection on a fractal billiard table that connects two elusive points of the fractal billiard table. The definition of a fractal flat surface is still not concretely stated, but in specific instances can be formally defined.\u00a0 In addition to such a surface, we have defined a fractal interval exchange transformation for the T-fractal flat surface.<\/span><\/p>\n<p style=\"padding-left: 40px\"><span style=\"font-size: 14pt\">My joint work with\u00a0<a href=\"http:\/\/homepages.uconn.edu\/jpchen\/\">Joe P. Chen<\/a>\u00a0on a project that extends the work of Jeremy Tyson and Estibalitz Durand-Cartagena\u00a0<a href=\"http:\/\/www.math.uiuc.edu\/~tyson\/papers\/carpets-lines.pdf\" target=\"_blank\" rel=\"noopener noreferrer\">[Du-CaTy]<\/a>\u00a0has recently appeared in the Journal of Mathematical Analysis &amp; Applications. The project involves describing periodic orbits of a self-similar Sierpinski carpet billiard table.<\/span><\/p>\n<p style=\"padding-left: 40px\"><span style=\"font-size: 14pt\">I have worked with\u00a0<a href=\"http:\/\/www.math.ucr.edu\/~lapidus\">M. L. Lapidus<\/a> on various papers on the topic of fractal billiards. Specifically, an article with M. L. Lapidus and <a href=\"http:\/\/mialab.mrn.org\/lab-members\/robyn-miller.html\">R. L. Miller<\/a>\u00a0consists of determining periodic orbits of the T-fractal billiard table. An eventual goal is understanding whether or not there are equidistributed orbits on the T-fractal billiard table. An initial step in this direction is to understand whether or not every orbit with an irrational direction is recurrent in the T-fractal billiard.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>C. Johnson and I are currently investigating the flow on a fractal flat surface. Related to this is another joint project concerning the existence of a saddle connection on a fractal billiard table that connects two elusive points of the fractal billiard table. The definition of a fractal flat surface is still not concretely stated, &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/sites.msudenver.edu\/niemeye1\/research-activities\/fractal-billiards\/\">Continue reading<\/a><\/p>\n","protected":false},"author":844,"featured_media":0,"parent":12,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_exactmetrics_skip_tracking":false,"_exactmetrics_sitenote_active":false,"_exactmetrics_sitenote_note":"","_exactmetrics_sitenote_category":0,"footnotes":""},"class_list":["post-49","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/sites.msudenver.edu\/niemeye1\/wp-json\/wp\/v2\/pages\/49","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.msudenver.edu\/niemeye1\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.msudenver.edu\/niemeye1\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.msudenver.edu\/niemeye1\/wp-json\/wp\/v2\/users\/844"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.msudenver.edu\/niemeye1\/wp-json\/wp\/v2\/comments?post=49"}],"version-history":[{"count":0,"href":"https:\/\/sites.msudenver.edu\/niemeye1\/wp-json\/wp\/v2\/pages\/49\/revisions"}],"up":[{"embeddable":true,"href":"https:\/\/sites.msudenver.edu\/niemeye1\/wp-json\/wp\/v2\/pages\/12"}],"wp:attachment":[{"href":"https:\/\/sites.msudenver.edu\/niemeye1\/wp-json\/wp\/v2\/media?parent=49"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}