{"id":3541,"date":"2023-02-13T14:57:18","date_gmt":"2023-02-13T19:57:18","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/boyland\/?page_id=3541"},"modified":"2026-03-19T08:10:24","modified_gmt":"2026-03-19T12:10:24","slug":"marlies-gerber","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/boyland\/marlies-gerber\/","title":{"rendered":"Marlies Gerber"},"content":{"rendered":"\r\n<section class=\"fullwidth-text-block\">\r\n\t<div class=\"container px-0 pt-5\">\r\n\t\t<div class=\"row align-items-start\">\r\n\t\t\t<div class=\"col-12\">\r\n\t\t\t\t\n<h1 class=\"wp-block-heading\">Marlies Gerber<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Complexity of Classification Problems For Dynamical Systems<\/h3>\n\n\n\n<p>Abstract:<\/p>\n\n\n\n<p>A classical problem in ergodic theory, posed by J. von Neumann in 1932, is the isomorphism problem: classify measure-preserving transformations up to conjugacy. Two great successes are the Halmos-von Neumann classification of ergodic transformations with pure-point spectrum in 1942 and Ornstein&#8217;s classification of Bernoulli shifts by their metric entropy in 1970. However, in 2001, Hjorth proved in a precise way that the isomorphism problem for general measure-preserving transformations is intractable, and in 2011, Foreman-Rudolph-Weiss showed that this is still true when the isomorphism problem is restricted to<\/p>\n\n\n\n<i>ergodic<\/i>\n\n\n\n<p>measure-preserving transformations. Nonetheless, it seemed feasible that the isomorphism problem might be solvable when restricted to families of measure-preserving transformations more general than Bernoulli shifts that have sufficiently strong mixing properties.<\/p>\n\n\n\n<p>We consider the isomorphism problem restricted to K-automorphisms. Within the collection of measure-preserving transformations, Bernoulli shifts have the ultimate mixing property, and K-automorphisms have the next-strongest mixing properties of any widely considered family of transformations. In particular, K-automorphisms have positive entropy and are mixing of all orders. It is known that, unlike Bernoulli shifts, the family of K-automorphisms cannot be classified up to isomorphism by a complete numerical Borel invariant. This left open the possibility of classifying\u00a0 K-automorphisms with a more complex type of Borel invariant. We show that this is impossible, by proving that the isomorphism equivalence relation restricted to K-automorphisms is complete analytic, and hence not Borel.<\/p>\n\n\n\n<p>Our work is primarily in the context of measurable dynamics, but I will also mention anti-classification results due to Foreman-Weiss, Foreman-Gorodetski, and Kunde in topological and differentiable dynamics.<\/p>\n\n\n\n<p>This talk is based on joint work with Philipp Kunde.<\/p>\n\n\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>\r\n<\/section>\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":146,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-3541","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/3541","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/users\/146"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/comments?post=3541"}],"version-history":[{"count":3,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/3541\/revisions"}],"predecessor-version":[{"id":3812,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/3541\/revisions\/3812"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/media?parent=3541"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}