Baker's Transformation

Stajner, Ivanka

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https://hdl.handle.net/2142/86948 Copy

Description

Title

Baker's Transformation

Author(s)

Stajner, Ivanka

Issue Date

1997

Doctoral Committee Chair(s)

Julian I. Palmore

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

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.).

Graduation Semester

1997

Type of Resource

text

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http://hdl.handle.net/2142/86948

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