{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86948"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86948","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Baker's Transformation","abstract":"101 p.","abstract_html":"101 p.","abstract_has_math":false,"creators":["Stajner, Ivanka"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Julian I. Palmore"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:20:18Z","date_published":"2015-09-28T15:20:18Z","updated_at":"2026-07-22T22:26:28Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI9737260"],"render_values":[{"text":"(MiAaPQ)AAI9737260","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86948","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Julian I. Palmore"]},{"key":"dc:creator","label":"Author","values":["Stajner, Ivanka"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-09-28T15:20:18Z","10000-01-01","1997"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/86948","(MiAaPQ)AAI9737260"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["101 p.","Dynamical properties of the baker's transformation B with integer base $b\\ge 2$ and several related maps on discrete subsets of the domain are studied. The baker's transformation with base $b,\\ B: \\lbrack 0,1)\\times\\lbrack 0,1)\\to\\lbrack 0,1) \\times \\lbrack 0,1)$ is defined by $B(x,y) = (S(x),\\ b\\sp{-1}\\ (y + bx - S(x))),$ where $S:\\lbrack 0,1)\\to\\lbrack 0,1)$ is the base b-shift given by $S(x) = bx (\\rm mod 1).$ For n any positive integer let $L\\sb{n} = \\{i/n: 0\\le i\\le n - 1\\},$ let $L\\sbsp{n}{\\*} = L\\sb{n} - \\{0\\},$ and let $\\overline{L}\\sb{n} = L\\sb{n}\\cup\\{1\\}.$ For p any prime that does not divide b, it is shown that all the orbits of the points on $L\\sbsp{p}{\\*}\\ \\times\\ L\\sb{p}$ are either periodic or attracted to periodic orbits, and all the periodic orbits have a common period. A constructive proof is given of the fact that the set of all periodic points of B is dense in $\\lbrack 0,1)\\times\\lbrack 0,1).$ For any positive integer n, recurrence properties of elementary n-squares, that is squares $I\\sb{n}(i,j) = \\lbrack i/b\\sp{n}, (i + 1)/b\\sp{n})\\times\\lbrack j/b\\sp{n}, (j + 1)/b\\sp{n})$ for integers i and j, $0\\le i,j\\le b\\sp{n} - 1,$ are studied. An upper bound of 2n is found for the smallest positive iterate $k(n; i, j)$ of B such that $B\\sp{k(n;i,j)}(I\\sb{n}(i,j))\\cap I\\sb{n}(i,j)\\not=\\emptyset.$ The number of squares $I\\sb{n}(i,j)$ such that $k(n; i, j) = k$ is less than the number P(k) of periodic points of B of period k for $n + 2\\le k \\le 2n,$ and it equals P(k) for $2\\le k\\le n + 1.$ A first integral for B on the lattice $L\\sb{n}\\times L\\sb{n}$ is shown to be $\\Phi\\sb{n}(i/n,j/n) = ij$ (mod n). The extended baker's transformation $\\overline{B}: \\lbrack 0,1)\\times\\lbrack 0,1)\\to\\lbrack 0,1)\\times\\lbrack 0,1\\rbrack$ is defined by $\\overline{B}(x,y) = (S(x), b\\sp{-1}(y + bx - S(x))).$ Let n be any positive integer. Rounding down $C\\sb{d},$ rounding up $C\\sb{u},$ and rounding to the nearest $C\\sb{m}$ from $\\lbrack 0,1\\rbrack\\times\\lbrack 0,1\\rbrack$ to $\\overline{L\\sb{n}}\\times\\overline{L\\sb{n}}$ are defined by $C\\sb{\\gamma}(x,y) = \\left({i\\over n},{j\\over n}\\right)$ where $i,j\\in \\{0,1,\\... n\\}$ and ${i\\over n}\\le x < {i+1\\over n}$ and ${j\\over n}\\le y < {j+1\\over n}$ for $\\gamma = d;\\ {i-1\\over n} < x \\le {i\\over n}$ and ${j-1\\over n} < y \\le {j\\over n}$ for $\\gamma = u;\\ {2i-1\\over 2n} \\le x < {2i+1\\over 2n}$ and ${2j-1\\over 2n}\\le y < {2j+1\\over 2n}$ for $\\gamma = m.$ It is shown that the orbit of a point on $L\\sb{n}\\times L\\sb{n}$ under either $C\\sb{d}\\circ B$ or $C\\sb{m}\\circ\\overline{B}$ is a 1/n-pseudo-orbit for $\\overline{B}$ and it is 1/n-shadowed by the orbit of the same point under $\\overline{B}.$ The conjugacy between $C\\sb{u}\\circ\\overline{B}$ and $C\\sb{d}\\circ\\overline{B}$ is exhibited. (Abstract shortened by UMI.).","Made available in DSpace on 2015-09-28T15:20:18Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9737260.pdf: 3555285 bytes, checksum: 0fb11d74b5a80db5d3d41dea2f7f99e1 (MD5) Previous issue date: 1997","Embargo set by: Seth Robbins for item 88229 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1997."]},{"key":"dc:title","label":"Title","values":["Baker's Transformation"]}]}],"canonical_facts":{"dc:contributor":["Julian I. Palmore"],"dc:creator":["Stajner, Ivanka"],"dc:date":["2015-09-28T15:20:18Z","10000-01-01","1997"],"dc:description":["101 p.","Dynamical properties of the baker's transformation B with integer base $b\\ge 2$ and several related maps on discrete subsets of the domain are studied. The baker's transformation with base $b,\\ B: \\lbrack 0,1)\\times\\lbrack 0,1)\\to\\lbrack 0,1) \\times \\lbrack 0,1)$ is defined by $B(x,y) = (S(x),\\ b\\sp{-1}\\ (y + bx - S(x))),$ where $S:\\lbrack 0,1)\\to\\lbrack 0,1)$ is the base b-shift given by $S(x) = bx (\\rm mod 1).$ For n any positive integer let $L\\sb{n} = \\{i/n: 0\\le i\\le n - 1\\},$ let $L\\sbsp{n}{\\*} = L\\sb{n} - \\{0\\},$ and let $\\overline{L}\\sb{n} = L\\sb{n}\\cup\\{1\\}.$ For p any prime that does not divide b, it is shown that all the orbits of the points on $L\\sbsp{p}{\\*}\\ \\times\\ L\\sb{p}$ are either periodic or attracted to periodic orbits, and all the periodic orbits have a common period. A constructive proof is given of the fact that the set of all periodic points of B is dense in $\\lbrack 0,1)\\times\\lbrack 0,1).$ For any positive integer n, recurrence properties of elementary n-squares, that is squares $I\\sb{n}(i,j) = \\lbrack i/b\\sp{n}, (i + 1)/b\\sp{n})\\times\\lbrack j/b\\sp{n}, (j + 1)/b\\sp{n})$ for integers i and j, $0\\le i,j\\le b\\sp{n} - 1,$ are studied. An upper bound of 2n is found for the smallest positive iterate $k(n; i, j)$ of B such that $B\\sp{k(n;i,j)}(I\\sb{n}(i,j))\\cap I\\sb{n}(i,j)\\not=\\emptyset.$ The number of squares $I\\sb{n}(i,j)$ such that $k(n; i, j) = k$ is less than the number P(k) of periodic points of B of period k for $n + 2\\le k \\le 2n,$ and it equals P(k) for $2\\le k\\le n + 1.$ A first integral for B on the lattice $L\\sb{n}\\times L\\sb{n}$ is shown to be $\\Phi\\sb{n}(i/n,j/n) = ij$ (mod n). The extended baker's transformation $\\overline{B}: \\lbrack 0,1)\\times\\lbrack 0,1)\\to\\lbrack 0,1)\\times\\lbrack 0,1\\rbrack$ is defined by $\\overline{B}(x,y) = (S(x), b\\sp{-1}(y + bx - S(x))).$ Let n be any positive integer. Rounding down $C\\sb{d},$ rounding up $C\\sb{u},$ and rounding to the nearest $C\\sb{m}$ from $\\lbrack 0,1\\rbrack\\times\\lbrack 0,1\\rbrack$ to $\\overline{L\\sb{n}}\\times\\overline{L\\sb{n}}$ are defined by $C\\sb{\\gamma}(x,y) = \\left({i\\over n},{j\\over n}\\right)$ where $i,j\\in \\{0,1,\\... n\\}$ and ${i\\over n}\\le x < {i+1\\over n}$ and ${j\\over n}\\le y < {j+1\\over n}$ for $\\gamma = d;\\ {i-1\\over n} < x \\le {i\\over n}$ and ${j-1\\over n} < y \\le {j\\over n}$ for $\\gamma = u;\\ {2i-1\\over 2n} \\le x < {2i+1\\over 2n}$ and ${2j-1\\over 2n}\\le y < {2j+1\\over 2n}$ for $\\gamma = m.$ It is shown that the orbit of a point on $L\\sb{n}\\times L\\sb{n}$ under either $C\\sb{d}\\circ B$ or $C\\sb{m}\\circ\\overline{B}$ is a 1/n-pseudo-orbit for $\\overline{B}$ and it is 1/n-shadowed by the orbit of the same point under $\\overline{B}.$ The conjugacy between $C\\sb{u}\\circ\\overline{B}$ and $C\\sb{d}\\circ\\overline{B}$ is exhibited. (Abstract shortened by UMI.).","Made available in DSpace on 2015-09-28T15:20:18Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 9737260.pdf: 3555285 bytes, checksum: 0fb11d74b5a80db5d3d41dea2f7f99e1 (MD5) Previous issue date: 1997","Embargo set by: Seth Robbins for item 88229 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1997."],"dc:identifier":["http://hdl.handle.net/2142/86948","(MiAaPQ)AAI9737260"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Baker's Transformation"],"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":"2026-07-22T22:26:28Z"}