Integer multiplication map on the circle ($T_K(x)=Kx(\mathrm{mod}1)$)

For an integer $K\ge 2$, multiplication by $K$ on the circle group preserves normalized Lebesgue measure. Each inverse image consists of $K$ branches, each scaling lengths by $1/K$.

The integer multiplication map on the circle is an ergodic transformation for Lebesgue measure. In the Fourier basis, an invariant function has coefficients $c_j=0$ when $K$ does not divide $j$, and $c_j=c_{j/K}$ otherwise. Every nonzero index eventually reduces to the first case, so only the constant coefficient remains.