Consider an interval exchange transformation that is, a bijective piecewise continuous map $[0,1] \rightarrow [0,1] $ whose restriction to its continuity intervals are translations. Assume that it is minimal and non uniquely ergodic.
Do the ergodic probability measures necessarily give the same weight to each of its continuity intervals?
I would naively expect the answer to this question to be negative, but since I have no example to test it
Thank you all!
Edit As Uri Bader pointed out in the comments, conjugating by a well chosen IET, any minimal non-uniquely ergodic IET provides a negative answer to the question, but the price to pay is to increase the number of continuity intervals involved. The 'good' question concerns then the IETs whose number of interval is minimal in its conjugacy class.