We will now apply the machinery already developed to a few concrete problems.

Proposition 1 Let {G} be a compact abelian group and {T} the rotation by {a \in G}. Then {T} is uniquely ergodic (with the Haar measure invariant) if {a^{\mathbb{Z}}} is dense in {G}.

 

The proof is straightforward. Suppose {\mu} is invariant with respect to rotations by {a}. Then for {f \in C(G)}, we have

\displaystyle \int f(a^m x ) d \mu = \int f(x) d \mu, \quad \forall m \in \mathbb{Z}

and hence

\displaystyle \int f(bx ) d \mu = \int f(x) d \mu, \quad \forall m \in \mathbb{Z},

for any {b \in G}, which means that {\mu} must be Haar measure (which is unique).

Corollary 2 An irrational rotation of the unit circle {S^1} is uniquely ergodic.

 

Application: Equidistribution

 

Theorem 3 Let {\xi \in \mathbb{R}} be irrational and let {f: \mathbb{R} \rightarrow \mathbb{C}} be continuous and {2 \pi }-periodic. Then\displaystyle \boxed{ \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{i=0}^{N-1} f( n \xi) = \int_0^1 f(x) dx .} (more…)