2026-09-21T18:01:22Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/26363 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Solecki, Slawomir Henson, C. Ward Solecki, Slawomir van den Dries, Lou Rosendal, Christian Hill, Aaron 2011-08-26T15:33:05Z 2013-08-27T10:00:25Z 2011-08-26T15:33:05Z 2011-08 In this dissertation we investigate centralizers in several automorphism groups of homogenous structures. In the rst chapter, we discuss the centralizer question in ergodic theory, an open question that has served as motivation for much of the work in this dissertation. We also introduce the content of each of the subsequent chapters and describe how it relates to the centralizer question in ergodic theory. In the second chapter, we investigate the topological complexity of the set of n-th powers in the group of isometries of Baire space. We prove that for n > 1, this set is not Borel. In the third chapter, we investigate topological similarity, an equivalence re- lation on a Polish group introduced by Rosendal in [15]. We prove some results for topological similarity in general Polish groups and give some new, simpli ed proofs of known genericity results in the group of invertible measure-preserving transformations. We also show that a generic measure-preserving transformation is not conjugate to any of its n-th roots, for n > 1. In the fourth chapter, we introduce the notion of a rank-1 homeomorphism of a zero-dimensional Polish space X, analogous in many ways to a rank-1 invertible measure-preserving transformation. We show that every rank-1 homeomorphism with a non-repeating tower representation (the class of such homeomorphisms is large) has trivial centralizer in the group of homeomorphisms of X. Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-07-04T18:09:57Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Hill_Aaron.pdf: 273658 bytes, checksum: eb6d4755ce955352c4ecb5fdfab7c961 (MD5) Made available in DSpace on 2011-08-26T15:33:05Z (GMT). No. of bitstreams: 2 Hill_Aaron.pdf: 273658 bytes, checksum: eb6d4755ce955352c4ecb5fdfab7c961 (MD5) license.txt: 4059 bytes, checksum: 7ccdca1c1db556f9d59cb2b9ccd6b6b4 (MD5) Item marked as restricted to the 'Administrator' Group (id=1) by William Ingram (wingram2@illinois.edu) on 2011-08-26T15:35:30Z Item is restricted until 2013-08-26T15:35:26Z Item reinstated by Sarah Shreeves (sshreeve@illinois.edu) on 2013-08-27T10:00:25Z Item was in collections: University of Illinois Dissertations and Theses (ID: 204) Dissertations and Theses - Mathematics (ID: 749) No. of bitstreams: 3 Hill_Aaron.pdf.txt: 89933 bytes, checksum: 68d3325a10c41db943b6168eb21c19a0 (MD5) Hill_Aaron.pdf: 273658 bytes, checksum: eb6d4755ce955352c4ecb5fdfab7c961 (MD5) license.txt: 4059 bytes, checksum: 7ccdca1c1db556f9d59cb2b9ccd6b6b4 (MD5) Item released from any restrictions by Sarah Shreeves (sshreeve@illinois.edu) on 2013-08-27T10:00:25Z http://hdl.handle.net/2142/26363 en Copyright 2011 Aaron Hill Rank-1 Topological Complexity Centralizer Centralizers in automorphism groups Mathematics 1257 Mathematics 0439 University of Illinois at Urbana-Champaign Dissertation Ph.D. PHD:Mathematics -UIUC 10KS0439PHD