TITLE: Minimal, uniquely ergodic but not Lebesgue-ergodic?
QUESTION [11 upvotes]: 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!

REPLY [2 votes]: I couldn't manage to find an online version, but this paper by Yoccoz provides an example of a diffeomorphism of the $2$-dimensional torus which is the product of two analytic circle diffeomorphisms, which is minimal, uniquely ergodic, and totally dissipative for Lebesgue measure (hence it cannot be ergodic).