Lebesgue-measure-preserving map ($S_{\ast }\mathrm{Leb}=\mathrm{Leb}$)

Such a measurable self-map preserves Lebesgue measure. On the unit interval examples include the identity, reflection $t\mapsto 1-t$, and $t\mapsto 2t(\mathrm{mod}1)$. Invertibility is not required.