So here's my question:

Does there exist a minimal diffeomorphism of class at least $\mathcal{C^2}$ of a compact manifold X which is

minimal

uniquely ergodic with unique probability measure $\mu$

not ergodic with respect to the Lebesgue measure ?

I don't really see why these requirements should contradict each other but I haven't been able to find an example. Note that the regularity hypothesis is necessary as (see R.W.'s answer below): there are $\mathcal{C}^1$ circle diffeomorphisms that satisfy those conditions, but one could argue that they are a bit artificial since as soon as the derivative is required to have bounded variation this can no longer be true.

I would also be happy with any example that is just a piecewise diffeomorphism!

class(one shouldn't really talk abouttheLebesgue measure on a smooth manifold) is quasi-invariant with respect to any diffeomorphism. $\endgroup$9more comments