We assume that $\frac { \sqrt { \lambda _ { 1 } } } { \sqrt { \lambda _ { 2 } } }$ is not a rational number. We denote by $C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ the set of functions $f \in C _ { 2 \pi , 2 \pi } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$ such that the two partial derivatives $\frac { \partial f } { \partial \theta _ { 1 } } , \frac { \partial f } { \partial \theta _ { 2 } }$ exist at every point of $\mathbb { R } ^ { 2 }$ and both define continuous functions on $\mathbb { R } ^ { 2 }$.
We set $\forall t \in \left[ 0 , + \infty \left[ , \theta ( t ) = \left( t \sqrt { \lambda _ { 1 } } + \varphi _ { 1 } , t \sqrt { \lambda _ { 2 } } + \varphi _ { 2 } \right) \right. \right.$.
The Ergodic Theorem states: Let $f \in C _ { 2 \pi , 2 \pi } ^ { 1 } \left( \mathbb { R } ^ { 2 } ; \mathbb { C } \right)$. Then, $$\lim _ { T \rightarrow + \infty } \frac { 1 } { T } \int _ { 0 } ^ { T } f \circ \theta ( t ) d t = ( 2 \pi ) ^ { - 2 } \int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } f \left( \theta _ { 1 } , \theta _ { 2 } \right) d \theta _ { 1 } d \theta _ { 2 } \tag{4}$$
Let $j , l \in \mathbb { Z }$. Prove the Ergodic Theorem in the special case of the function $\left( \theta _ { 1 } , \theta _ { 2 } \right) \mapsto f \left( \theta _ { 1 } , \theta _ { 2 } \right) = e ^ { i \theta _ { 1 } j } e ^ { i \theta _ { 2 } l }$. (In the case where $( j , l ) \neq ( 0,0 )$ one may verify that $j \sqrt { \lambda _ { 1 } } + l \sqrt { \lambda _ { 2 } }$ is non-zero and then one may calculate each side of (4) separately in this special case).