Uniquely Ergodic Transformation on compact groups

Question

I am currently reading Einsiedler & Ward's book Ergodic Theory with a view towards Number Theory (GTM259) and am stuck at one of the exercises:

Exercise 4.3.2 (p.109): Let $T: X \to X$ be a surjective homomorphism on a compact group $X$. Then $T$ is uniquely ergodic iff $|X| = 1$.

Any help or hint would be highly appreciated.

Answer 1

The Dirac function $\delta_e$ at the origin is invariant by a morphism of a compact group. Its support is the set $\{e\}$, where $e$ is the identity element of the group.

A compact group admits a unique left invariant probability measure, called the Haar measure. You can check that it is invariant by any surjective morphism. The Haar measure is of full support.

If a surjective morphism is uniquely ergodic, then these two measures are equal. So are their supports. Hence $X$ is equal to $\{e\}$.