Prove that the rotation map $T : \Bbb T → \Bbb T$ defined as $T(x) = x + 3/4$ mod $1$ is not ergodic

Question

Prove that the rotation map $T : \Bbb T → \Bbb T$ defined as $T(x) = x + 3/4$ mod $1$ is not ergodic where $\Bbb T=[0,1]/\{0=1\}$. I know that a map would be ergodic if $T^{-1}E=E$ would imply that $m(E)=0$ or $1$. So I think that I have to violate that.

Answer 1

Yes, you are right.

Check that the set $[0, 1/8) ∪ [1/4, 3/8) ∪ [1/2, 5/8) ∪ [3/4, 7/8)$ is an invariant set under $T$. Since it has measure $\frac 12$, Hence, $T$ is not ergodic.