Baker's Map - Formal Definition

Formal Definition

There are two alternative definitions of the baker's map which are in common use. One definition folds over or rotates one of the sliced halves before joining it (similar to the horseshoe map) and the other does not.

The folded baker's map acts on the unit square as

$S_\text{baker-folded}(x, y) = \begin{cases} (2x, y/2) & \text{for } 0 \le x < \frac{1}{2} \\ (2-2x, 1-y/2) & \text{for } \frac{1}{2} \le x < 1. \end{cases}$

When the upper section is not folded over, the map may be written as

$S_\text{baker-unfolded}(x,y)= \left(2x-\left\lfloor 2x\right\rfloor \,,\,\frac{y+\left\lfloor 2x\right\rfloor }{2}\right).$

The folded baker's map is a two-dimensional analog of the tent map

$S_\mathrm{tent}(x) = \begin{cases} 2x & \text{for } 0 \le x < \frac{1}{2} \\ 2(1-x) & \text{for } \frac{1}{2} \le x < 1 \end{cases}$

while the unfolded map is analogous to the Bernoulli map. Both maps are topologically conjugate. The Bernoulli map can be understood as the map that progressively lops digits off the dyadic expansion of x. Unlike the tent map, the baker's map is invertible.

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