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 


1. minimal
2. uniquely ergodic with unique probability measure $\mu$
3. 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!



Minimal, uniquely ergodic but not Lebesgue-ergodic?
I couldn't manage to find an online version, but [this paper](https://mathscinet.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=yoccoz&s5=denjoy-koksma&s6=&s7=&s8=All&sort=Newest&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq) 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).


