{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/105632"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/105632","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Geometric and ergodic properties of certain classes of continued fractions","abstract":"Made available in DSpace on 2019-11-26T20:33:47Z (GMT). No. of bitstreams: 2 MERRIMAN-DISSERTATION-2019.pdf: 3545539 bytes, checksum: 1ceb6dccbf49a654427e8d1f15d2be3d (MD5) LICENSE.txt: 4212 bytes, checksum: b6441bf949f57dfb6f36245b79c18128 (MD5) Previous issue date: 2019-07-08","abstract_html":"Made available in DSpace on 2019-11-26T20:33:47Z (GMT). No. of bitstreams: 2 MERRIMAN-DISSERTATION-2019.pdf: 3545539 bytes, checksum: 1ceb6dccbf49a654427e8d1f15d2be3d (MD5) LICENSE.txt: 4212 bytes, checksum: b6441bf949f57dfb6f36245b79c18128 (MD5) Previous issue date: 2019-07-08","abstract_has_math":false,"creators":["Merriman, Claire"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Boca, Florin P","Leininger, Christopher","Tyson, Jeremy","Zaharescu, Alexandru"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-11-26T20:33:47Z","date_published":"2019-11-26T20:33:47Z","updated_at":"2026-07-22T22:24:44Z","subjects":["ergodic theory, continued fractions, cutting sequence"],"languages":["en"],"rights":["Copyright 2019 Claire Merriman"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/105632","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Boca, Florin P","Leininger, Christopher","Tyson, Jeremy","Zaharescu, Alexandru"]},{"key":"dc:creator","label":"Author","values":["Merriman, Claire"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2019-11-26T20:33:47Z","2019-07-08","2019-08"]},{"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":["ergodic theory, continued fractions, cutting sequence"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2019 Claire Merriman"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/105632"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Made available in DSpace on 2019-11-26T20:33:47Z (GMT). No. of bitstreams: 2 MERRIMAN-DISSERTATION-2019.pdf: 3545539 bytes, checksum: 1ceb6dccbf49a654427e8d1f15d2be3d (MD5) LICENSE.txt: 4212 bytes, checksum: b6441bf949f57dfb6f36245b79c18128 (MD5) Previous issue date: 2019-07-08","Continued fractions are systematically studied in number theory, dynamical systems, and ergodic theory, and appear frequently in other areas of mathematics. The first part of this thesis extends the connection between the dynamics of regular continued fractions and the geodesics on the modular surface $\\operatorname{PSL}(2, \\mathbb{Z})\\backslash\\mathbb{H}$ to the odd and grotesque continued fractions and the even continued fractions, as well as the Lehner and Farey expansions. I describe the natural extension of the corresponding Gauss maps as cross sections of the geodesic flow on modular surfaces. For the odd and grotesque continued fractions, $\\Gamma$ is an index two subgroup of $\\operatorname{PSL}(2,\\Z)$; for the even continued fractions, $\\Gamma$ is an index three subgroup; and for the Lehner and Farey expansions, $\\Gamma=\\operatorname{PSL}(2,\\Z)$. Nakada’s $\\alpha$-expansions interpolate between the regular continued fractions ($\\alpha=1$), Hurwitz singular continued fractions ($\\alpha=\\frac{\\sqrt{5}-1}{2}$), and nearest integer continued fractions ($\\alpha=\\frac{1}{2}$). The second part of this thesis introduces a new version of $\\alpha$-expansions where all partial quotients are odd integers. I provide an explicit description of the natural extension of the corresponding Gauss map for $\\frac{\\sqrt{5}-1}{2}\\leq\\alpha\\leq\\frac{\\sqrt{5}+1}{2}$, and investigate several of the ergodic properties of these maps.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-11-26 without embargo terms","The student, Claire Merriman, accepted the attached license on 2019-07-01 at 16:41.","The student, Claire Merriman, submitted this Dissertation for approval on 2019-07-01 at 16:48.","This Dissertation was approved for publication on 2019-07-08 at 11:18.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14126 on 2019-11-26 at 12:50:23"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Geometric and ergodic properties of certain classes of continued fractions"]}]}],"canonical_facts":{"dc:contributor":["Boca, Florin P","Leininger, Christopher","Tyson, Jeremy","Zaharescu, Alexandru"],"dc:creator":["Merriman, Claire"],"dc:date":["2019-11-26T20:33:47Z","2019-07-08","2019-08"],"dc:description":["Made available in DSpace on 2019-11-26T20:33:47Z (GMT). No. of bitstreams: 2 MERRIMAN-DISSERTATION-2019.pdf: 3545539 bytes, checksum: 1ceb6dccbf49a654427e8d1f15d2be3d (MD5) LICENSE.txt: 4212 bytes, checksum: b6441bf949f57dfb6f36245b79c18128 (MD5) Previous issue date: 2019-07-08","Continued fractions are systematically studied in number theory, dynamical systems, and ergodic theory, and appear frequently in other areas of mathematics. The first part of this thesis extends the connection between the dynamics of regular continued fractions and the geodesics on the modular surface $\\operatorname{PSL}(2, \\mathbb{Z})\\backslash\\mathbb{H}$ to the odd and grotesque continued fractions and the even continued fractions, as well as the Lehner and Farey expansions. I describe the natural extension of the corresponding Gauss maps as cross sections of the geodesic flow on modular surfaces. For the odd and grotesque continued fractions, $\\Gamma$ is an index two subgroup of $\\operatorname{PSL}(2,\\Z)$; for the even continued fractions, $\\Gamma$ is an index three subgroup; and for the Lehner and Farey expansions, $\\Gamma=\\operatorname{PSL}(2,\\Z)$. Nakada’s $\\alpha$-expansions interpolate between the regular continued fractions ($\\alpha=1$), Hurwitz singular continued fractions ($\\alpha=\\frac{\\sqrt{5}-1}{2}$), and nearest integer continued fractions ($\\alpha=\\frac{1}{2}$). The second part of this thesis introduces a new version of $\\alpha$-expansions where all partial quotients are odd integers. I provide an explicit description of the natural extension of the corresponding Gauss map for $\\frac{\\sqrt{5}-1}{2}\\leq\\alpha\\leq\\frac{\\sqrt{5}+1}{2}$, and investigate several of the ergodic properties of these maps.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2019-11-26 without embargo terms","The student, Claire Merriman, accepted the attached license on 2019-07-01 at 16:41.","The student, Claire Merriman, submitted this Dissertation for approval on 2019-07-01 at 16:48.","This Dissertation was approved for publication on 2019-07-08 at 11:18.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14126 on 2019-11-26 at 12:50:23"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/105632"],"dc:language":["en"],"dc:rights":["Copyright 2019 Claire Merriman"],"dc:subject":["ergodic theory, continued fractions, cutting sequence"],"dc:title":["Geometric and ergodic properties of certain classes of continued fractions"],"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":"2026-07-22T22:24:44Z"}