Minimal, uniquely ergodic but not Lebesgue-ergodic? 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!
 A: You haven't specified the smoothness, so that hopefully $C^1$ is OK. It was Denjoy who proved in 1932 that if a $C^1$ diffeomorphism $f$ of the circle has an irrational rotation number $\alpha$ and its derivative has bounded variation, then it is $C^0$-conjugate to the $\alpha$-rotation, and therefore is uniquely ergodic. Answering Denjoy's question, Herman (1979) and Katok (see Section 3.6 of Cornfeld-Fomin-Sinai) proved that if $f$ is $C^2$ and an irrational rotation number, then it is also ergodic with respect to the Lebesgue measure. Later the $C^2$ condition was replaced by Katok-Hasselblatt with Denjoy's condition ($C^1$ and bounded variation of the derivative).  
Oliveira and da Rocha (2001) gave an example of a minimal non-ergodic (with respect to the Lebesgue measure) $C^1$ diffeomorphism $f$ of the circle which is $C^0$ conjugate to an irrational rotation (and therefore is uniquely ergodic). Finally, Kodama and Matsumoto (2013) showed that non-ergodicity in such examples can be made "the strongest possible", namely $f$ can be chosen to be completely dissipative with respect to the Lebesgue measure, i.e., such that its ergodic components are just orbits, or, equivalently, it admits a measurable "fundamental domain". 
A: 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).
