Baker transformation

The baker's map (English baker's map ) was named after the process of Teigknetens: A dough is pulled to twice the length and then folded. This procedure repeats itself until a good mix is ​​obtained. Two hypothetical particles in the dough (assumed to be point-like “raisins”), which were originally close to each other , are far apart after repeated use of this transformation .

With the baker's transformation, it is easy to illustrate how chaotic behavior arises from the interplay of stretches and folds .

Formal description

Various formulas for the baker's transformation can be found in the literature. The formulas differ by

• the type of folding as well
• viewing the dough as infinitely thin or of finite thickness.

Infinitely thin dough

The first 50 iterations of the baker's transformation (infinitely thin dough with folding back) for the initial value 0.0123456789

For the sake of simplicity, consider a one-dimensional dough (since the dough is only stretched in one direction, the second direction parallel to the table surface does not matter). Formally, we can represent this piece of dough by the unit interval . The Baker's transformation is then a mapping of the unit interval in itself, i.e. H. . ${\displaystyle x\in [0,1]}$ ${\displaystyle f\colon [0,1]\rightarrow [0,1]}$

The simplest form of the baker's transformation results when you roll out the dough to double its length and then fold it so that the two ends come to rest on top of each other. Mathematically, this transformation can be described as follows:

${\displaystyle x_{n+1}:=f(x_{n}):={\begin{cases}2x_{n}&{\mbox{falls }}x_{n}\in [0,1/2]\\2-2x_{n}&{\mbox{falls }}x_{n}\in (1/2,1]\end{cases}}}$

This figure is also known as a tent figure .

An alternative form of the baker's transformation is obtained by cutting the rolled out dough in half and placing the two halves on top of one another without turning them against each other. For this is the mathematical description

${\displaystyle x_{n+1}:=f(x_{n}):={\begin{cases}2x_{n}&{\mbox{falls }}x_{n}\in [0,1/2]\\2x_{n}-1&{\mbox{falls }}x_{n}\in (1/2,1]\end{cases}}}$

This mapping is also known as the Bernoulli mapping .

Dough of finite thickness

When looking at a dough of finite thickness, we need to introduce a second variable that describes the vertical distance of our hypothetical particle from the table top. The variables and are mostly used in the literature ; here the height is to be represented by the variable in order to enable a more intuitive understanding of the formulas. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$

For the sake of simplicity, consider a dough of height 1, so that the baker's transformation is now a representation of the unit square in itself, i.e. H. . The figure in Figure 13 is the same as above; the figure in is based on the consideration that the thickness of the dough is halved when rolling to double the length and from the respective folding operation. ${\displaystyle f\colon [0,1]^{2}\rightarrow [0,1]^{2}}$ ${\displaystyle x}$ ${\displaystyle z}$

If the dough is folded over after being rolled out, the mathematical description is ${\displaystyle (x_{n+1},z_{n+1}):=f(x_{n},z_{n}):={\begin{cases}(2x_{n},{\frac {z_{n}}{2}})&{\mbox{falls }}x_{n}\in [0,1/2]\\(2-2x_{n},1-{\frac {z_{n}}{2}})&{\mbox{falls }}x_{n}\in (1/2,1]\end{cases}}}$

If, on the other hand, the dough is cut through and laid on top of one another, one obtains ${\displaystyle (x_{n+1},z_{n+1}):=f(x_{n},z_{n}):={\begin{cases}(2x_{n},{\frac {z_{n}}{2}})&{\mbox{falls }}x_{n}\in [0,1/2]\\(2x_{n}-1,{\frac {1+z_{n}}{2}})&{\mbox{falls }}x_{n}\in (1/2,1]\end{cases}}}$

swell

• John H. Argyris, Gunter Faust, Maria Haase: The exploration of chaos . Vieweg, Braunschweig 1994, ISBN 3-528-08941-5 .
• Erich W. Weisstein: Baker's Map. From MathWorld - A Wolfram Web Resource ( see here )
• Roman Worg: Deterministic Chaos. Paths to nonlinear dynamics . BI-Wissenschaftsverlag, Mannheim 1993, ISBN 3-411-16251-1 .