Do ergodicity, minimality and equicontinuity on a compact space imply total ergodicity? 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.
 A: 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.
A: Just knowing that a transformation $T$ is minimal is no guarantee that $T^n$ is also minimal. For example, let $T_1$ be the non-identity homeomorphism of a two-point metric space and let let $T_2$ be an aperiodic, minimal, equicontinuous, uniquely ergodic transformation of a compact metric space (for example, an irrational circle rotation). Then $T_1 \times T_2$ is minimal and uniquely ergodic, but $T_1^2 \times T_2^2$ has two nonempty, disjoint, closed invariant sets, and the unique invariant measure for $T_1 \times T_2$ is not ergodic for $T_1^2 \times T_2^2$, since it gives both "halves" of the phase space positive measure, but those "halves" are invariant sets for $T_1^2 \times T_2^2$.
