Dyadic transformation

Example

The dyadic transformation, or doubling map, is $$T \colon [0, 1) \to [0, 1), \qquad T(x) = 2x \bmod 1.$$

It is a measure-preserving transformation for Lebesgue measure $\lambda$. Check on an interval $[a, b) \subseteq [0, 1)$: $$T^{-1}[a, b) = \Big[\frac{a}{2}, \frac{b}{2}\Big) \cup \Big[\frac{a+1}{2}, \frac{b+1}{2}\Big),$$ a disjoint union of two intervals of length $(b - a)/2$, so $\lambda(T^{-1}[a, b)) = b - a$.

Write $x = \sum_{n \ge 1} x_n 2^{-n}$ in binary, with digits $x_n \in \set{0, 1}$ not eventually all equal to $1$. Then $T$ deletes the first digit: $T(x) = \sum_{n \ge 1} x_{n+1} 2^{-n}$. Under $\lambda$ the digits are independent fair coin tosses. So the binary expansion is an isomorphism, up to null sets, from $T$ to the one-sided Bernoulli scheme $B(1/2, 1/2)$. Hence $T$ is mixing, and its Kolmogorov–Sinai entropy is $\log 2$.

The map is not invertible: each point has two preimages.