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Title:Baker's Transformation
Author(s):Stajner, Ivanka
Doctoral Committee Chair(s):Julian I. Palmore
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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.).
Issue Date:1997
Description:101 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1997.
Other Identifier(s):(MiAaPQ)AAI9737260
Date Available in IDEALS:2015-09-28
Date Deposited:1997

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