Baker's map

Example

Let $X = [0, 1) \times [0, 1)$ be the unit square with two-dimensional Lebesgue measure $\lambda$. The baker's map $B \colon X \to X$ is $B(x, y) = (2x, y/2)$ for $0 \le x < 1/2$. For $1/2 \le x < 1$, it is $B(x, y) = (2x - 1, (y + 1)/2)$. It stretches the square to twice its width and half its height. Then it cuts the result in two and stacks the right half on top of the left half, as a baker folds dough.

It is a bijection: the left half $[0, 1/2) \times [0, 1)$ goes onto the bottom half $[0, 1) \times [0, 1/2)$, and the right half goes onto the top half. On each half, $B$ is affine with Jacobian determinant $2 \cdot \frac{1}{2} = 1$, so $\lambda(B^{-1} E) = \lambda(E)$ for every Borel set $E$, and $B$ is a measure-preserving transformation.

Write $x = 0.x_1 x_2 x_3 \dots$ and $y = 0.y_1 y_2 y_3 \dots$ in binary, and code the point $(x, y)$ by the two-sided sequence $\dots y_2 y_1 . x_1 x_2 \dots$. Then $B$ moves the binary point one step to the right: the digit $x_1$ leaves $x$ and becomes the first digit of the new $y$. This coding is an isomorphism, up to null sets, from $B$ to the two-sided Bernoulli scheme $B(1/2, 1/2)$. Hence $B$ is invertible, mixing, and has Kolmogorov–Sinai entropy $\log 2$.