Show that $T^n(x,y)=\left(x+n\alpha \mod 1, y+nx+\frac{n(n-1)}{2}\alpha \mod 1 \right)$

Question

Let $\mathbb{T}^2=\mathbb{R}^2 / \mathbb{Z}^2$. Let $T: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be the transformation. Let $\alpha \in \mathbb{R}$.

$$T(x,y)=\left(x+\alpha \mod 1, x+y \mod 1 \right) $$.

Show that $T^n(x,y)=\left(x+n\alpha \mod 1, y+nx+\frac{n(n-1)}{2}\alpha \mod 1 \right)$.

I get that $T^n(x,y)=\left(x+n\alpha \mod 1, y+nx \mod 1 \right)$. I cannot see you could obtain $\frac{n(n-1)}{2}\alpha$. Clearly $\frac{n(n-1)}{2}$ is an integer as $n(n-1)$ will be even, but $\alpha$ could be anything???

Answer 1

Hint: Use induction. You should find yourself mimicking the proof that $$\sum_{i=1}^{n-1} i = \frac{n(n-1)}{2}.$$ You essentially have a multiplication by $\alpha$, which is no big deal since it can be factored out.