Definition
Total ergodicity
Ergodicity of every positive power of a probability-preserving transformation.
A probability-preserving transformation is totally ergodic if is ergodic for every integer .
Spectral criterion
For an invertible ergodic system, this is equivalent to the absence of Koopman eigenvalues that are nontrivial roots of unity. Indeed, such an eigenfunction becomes fixed under a power. Conversely, a nonconstant vector fixed by lies in a finite-dimensional invariant space generated by its first iterates, which has a root-of-unity eigenvector.
Irrational torus translations are totally ergodic. A single finite cycle of length greater than one is ergodic but not totally ergodic. Weak mixing implies total ergodicity, but irrational rotations show that the converse fails.