Consider the measure preserving dynamical system $(\mathbb{R}^2 / \mathbb{Z}^2, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda, R_{(\alpha, \beta)})$. This is the torus with the borel $\sigma$-algebra, the Lebesgue measure, and the rotation defines as
$$
R_{(\alpha, \beta)}(x,y) = (x + \alpha \ (mod1), y + \beta \ (mod1))
$$
I'm searching for a necessary and suficient condition for this system to be ergodic.
I have come by using Fourier Analysis to a sufficient condition for the system to be ergodic:
$$
\forall (k_1, k_2) \in \mathbb{Z}^2 \backslash \{(0,0)\}, \ k_1 \alpha + k_2 \beta \notin \mathbb{Z}
$$
And by a more geometric analysis to a necessary condition for the system to be ergodic:
$$
\forall \lambda > 0, \ \lambda \cdot (\alpha, \beta) \neq 0
$$
But I haven't been able to unify them into one.
Any help or good references on this topic?
Thanks in advance.
Best Answer
Note that the system is ergodic if and only if the orbit of each point is equidistributed in $\mathbb{R}^2 / \mathbb{Z}^2$.
The first condition $$\forall (k_1, k_2) \in \mathbb{Z}^2 \backslash \{(0,0)\}, \ k_1 \alpha + k_2 \beta \notin \mathbb{Z}$$ is a necessary and sufficient condition. You can use Weyl criterion for equidistribution to show this. The Weyl criterion says that a sequence $x:\mathbb{N}\to(\mathbb{R}/\mathbb{Z})^d$ is equidistributed in $(\mathbb{R}/\mathbb{Z})^d$ if and only if $$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}e(k\cdot x(n))=0 \text{ for all } k\in\mathbb{Z}^d\backslash\{0\},$$ where $e(y):=e^{2\pi iy}$ and $(k_1,\ldots,k_d)\cdot(x_1,\ldots,x_d)=k_1x_1+\ldots+k_dx_d$. It is not hard to see that the first condition is equivalent to the above formula.
For a reference, you can find the proof of Weyl criterion in Terence Tao's book "Higher order Fourier analysis".