Ergodic system counterexample

Question

Can anyone think of an easy example when $$(X, \mathcal{B}, \mu, T)$$ is an ergodic measure-preserving system but $$(X, \mathcal{B}, \mu, T^k)$$ is not ergodic for $k$ a positive integer.

Answer 1

What about the rotation by $\pi$ on the unit circle, i.e.

$(X, \mathcal{A}) = (\mathbb{S}^1, \mathcal{B}(\mathbb{S}^1)),$ where $\mathbb{S}^1 = \lbrace z \in \mathbb{C : |z| = 1} \rbrace$ and $\mathcal{B}(\mathbb{S}^1)$ is the corresponding Borel-$\sigma$-Algebra. Then define $$T(z) = e^{i\pi}z$$ for all $z \in \mathbb{S}^1$ and $$\mu = \frac{1}{2}(\delta_{-1} + \delta_{1})$$ as the equidistribution on $\lbrace -1, 1\rbrace$. Note that $T$ is measurable as a continuous transformation. It is even measure preserving because if a set $A \in \mathcal{B}(\mathbb{S}^1)$ contains $-1$, $T^{-1}(A)$ contains $1$ and the other way around. It is also ergodic because any set $A \in \mathcal{B}(\mathbb{S}^1)$ with $T^{-1}(A) = A$ needs to either contain $1$ and $-1$ ($\Rightarrow \mu(A) = 1$) or none of both ($\Rightarrow \mu(A) = 0).$

$T^2$ however is the identity which is not ergodic since $T^{-2}(\lbrace 1 \rbrace) = \lbrace 1 \rbrace$ and $\mu(\lbrace 1 \rbrace) = \frac{1}{2}$.