{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/21910"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/21910","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A global Boettcher's theorem","abstract":"We prove a global result for rational functions that is analogous to a local theorem of L. E. Boettcher (1904). Under the hypotheses that f is a complex rational function with a superattractive fixed point $\\alpha$ of order $p \\ge 2$ and that $\\alpha$ is the only critical point of f in the immediate basin of attraction of $\\alpha,$ we prove that there exists a conformal map $w = \\varphi(z)$ of the entire immediate basin of attraction of $\\alpha$ onto the unit disk such that $(\\varphi \\circ f \\circ \\varphi\\sp{-1})(w) = w\\sp{p}.$","abstract_html":"We prove a global result for rational functions that is analogous to a local theorem of L. E. Boettcher (1904). Under the hypotheses that f is a complex rational function with a superattractive fixed point <span class=\"etd-inline-math\">&alpha;</span> of order $p \\ge 2$ and that <span class=\"etd-inline-math\">&alpha;</span> is the only critical point of f in the immediate basin of attraction of <span class=\"etd-inline-math\">&alpha;,</span> we prove that there exists a conformal map $w = \\varphi(z)$ of the entire immediate basin of attraction of <span class=\"etd-inline-math\">&alpha;</span> onto the unit disk such that $(\\varphi \\circ f \\circ \\varphi\\sp{-1})(w) = w\\sp{p}.$","abstract_has_math":true,"creators":["Kline, Bradford J."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Miles, Joseph B."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:22:56Z","date_published":"2011-05-07T13:22:56Z","updated_at":"2026-07-22T22:25:18Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1995 Kline, Bradford J."],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9543631","(UMI)AAI9543631"],"render_values":[{"text":"AAI9543631","href":null,"code":true},{"text":"(UMI)AAI9543631","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/21910","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Miles, Joseph B."]},{"key":"dc:creator","label":"Author","values":["Kline, Bradford J."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:22:56Z","10000-01-01","1995"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1995 Kline, Bradford J."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9543631","(UMI)AAI9543631","http://hdl.handle.net/2142/21910"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We prove a global result for rational functions that is analogous to a local theorem of L. E. Boettcher (1904). Under the hypotheses that f is a complex rational function with a superattractive fixed point $\\alpha$ of order $p \\ge 2$ and that $\\alpha$ is the only critical point of f in the immediate basin of attraction of $\\alpha,$ we prove that there exists a conformal map $w = \\varphi(z)$ of the entire immediate basin of attraction of $\\alpha$ onto the unit disk such that $(\\varphi \\circ f \\circ \\varphi\\sp{-1})(w) = w\\sp{p}.$","In our proof, the conjugating function $\\varphi$ appears as the unique fixed point of a certain contraction operator on a complete metric space of one-to-one analytic functions. We make extensive use of the topological concept of a branched covering space in defining the contraction operator and, hence, in obtaining the global existence of $\\varphi.$","Made available in DSpace on 2011-05-07T13:22:56Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9543631.pdf: 1846358 bytes, checksum: 33e97bfc74cb318ca00469ce264701e8 (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:54:01Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:25:03-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["A global Boettcher's theorem"]}]}],"canonical_facts":{"dc:contributor":["Miles, Joseph B."],"dc:creator":["Kline, Bradford J."],"dc:date":["2011-05-07T13:22:56Z","10000-01-01","1995"],"dc:description":["We prove a global result for rational functions that is analogous to a local theorem of L. E. Boettcher (1904). Under the hypotheses that f is a complex rational function with a superattractive fixed point $\\alpha$ of order $p \\ge 2$ and that $\\alpha$ is the only critical point of f in the immediate basin of attraction of $\\alpha,$ we prove that there exists a conformal map $w = \\varphi(z)$ of the entire immediate basin of attraction of $\\alpha$ onto the unit disk such that $(\\varphi \\circ f \\circ \\varphi\\sp{-1})(w) = w\\sp{p}.$","In our proof, the conjugating function $\\varphi$ appears as the unique fixed point of a certain contraction operator on a complete metric space of one-to-one analytic functions. We make extensive use of the topological concept of a branched covering space in defining the contraction operator and, hence, in obtaining the global existence of $\\varphi.$","Made available in DSpace on 2011-05-07T13:22:56Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9543631.pdf: 1846358 bytes, checksum: 33e97bfc74cb318ca00469ce264701e8 (MD5) Previous issue date: 1995","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:54:01Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:25:03-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9543631","(UMI)AAI9543631","http://hdl.handle.net/2142/21910"],"dc:language":["eng"],"dc:rights":["Copyright 1995 Kline, Bradford J."],"dc:subject":["Mathematics"],"dc:title":["A global Boettcher's theorem"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:18Z"}