Doubling map of the circle

The doubling map is $D:\mathbb R/\mathbb Z\to\mathbb R/\mathbb Z$ , $D(x)=2x\bmod1$ , with Lebesgue probability. It preserves measure, is strongly mixing, and is not invertible modulo null sets.

Measure preservation

The two inverse branches $x\mapsto x/2$ and $x\mapsto(x+1)/2$ each halve lengths. Changing variables on the two half-intervals gives $\int f\circ D=\int f$ for integrable $f$ .

Koopman proof of mixing

The Fourier basis satisfies $U_De_k=e_{2k}$ , so $\langle U_D^ne_k,e_\ell\rangle=0$ eventually for fixed $k\ne0,\ell$ . Trigonometric approximation and the isometry bound extend this to mean-zero $L^2$ observables. This proves mixing.

The range of $U_D$ is the closed span of even frequencies, and omits $e_1$ . An invertible measure-space transformation would give a surjective Koopman operator, so $D$ is not invertible even modulo null sets.