I think a related question, which encapsulate all the difficulties of yours but lowers the dimension, is the following: can one find orbit closures of the doubling map $x \in [0,1[ \mapsto 2x \ mod \ 1$ of arbitrary (between 0 and 1) Hausdorff dimension?

By global time change do you mean conjugated to via a $\mathcal{C}^{\infty}$? Because if it is the case the answer to your question is no. It is formally what I have said in my previous comment (see arxiv.org/pdf/1003.1191.pdf for precise definitions). I think a good way to say it is that $\mathcal{C}^{\infty}$ interval exchange transformation are to standard interval exchange transformations what circle diffeos are to rotations.

A piecewise continuous bijection of the interval with finitely many discontinuity points which $\mathcal{C}^{\infty}$ where it is continuous. It is sometimes called 'generalised' interval exchange transformation.

Hi RW, thank you for your answer. Actually what I had in mind was the problem in high regularity as I was aware of those examples (sorry for not being precise enough in my question). What actually initiated my interest in this question is the following problem: can you find minimal, uniquely ergodic $\mathcal{C}^{\infty}$ interval exchange transformations that are not Lebesgue ergodic? which would be the natural generalisation of the Herman-Katok result you are referring to...

@ChristianRemling No I didn't meant that. Although item 2 and 3 together imply that the invariant measure is not equivalent to Lebesgue, 3. demands that Lebesgue is not ergodic which is much stronger.

@ChristianRemling I am making no assumption on the ergodic measure. It seems to me that the question makes sense formulated as it is...

In that case you won’t be able to produce any example.

Hi Christian. If you use "non-standard coordinates", then you loose the fact that your map is a diffeo. For any smooth manifold, the class of the Lebesgue measure is well-defined and it makes sense to say that a diffeo is ergodic (when the measure of any invariant set or its complement is zero for any representative of the class). It is standard that any sufficiently regular ($\mathcal{C}^2$) minimal circle diffeo is ergodic with respect to Lebesgue.

Hi Richard thank you for the nice answer. I need to reformulate Q3 because that is not exactly what I had in mind: I meant embedding in a way which is $\pi_1$-injective because otherwise indeed you can play the game you suggested.