Is it true than an aperiodic, ergodic, minimal and equicontinuous dynamical system on a compact metric space is totally ergodic ?


According to some results I found in some books, a rotation on a compact metric group is equicontinous, and it is minimal ~~and totally ergodic~~ whenever it is ergodic.



Do ergodicity, minimality and equicontinuity on a compact space imply total ergodicity?
The answer is no in this generality! If you consider the classical odometer (i.e. addition by 1 on 2-adic integers) then its second power (addition by 2) is not minimal. This second power preserves the numbers starting with 0 (the "even numbers") and the numbers starting with 1 (the "odd numbers"). But of course this example is totally disconnected.


