Is $T:(x,y)\mapsto(x+\alpha, y+x)$ mod $1$, expansive on $\mathbb{R}^2 / \mathbb{Z}^2$?

Question

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

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

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

DEFINITION: A topological dynamical system $f:X \rightarrow X$ (essentially just a continuous map) is called expansive if there exists $\nu > 0$, such that for all $x,y \in X$ such that $x \neq y$ there exists $n \in \mathbb{N}$ such that

$$d(f^n(x),f^n(y)) \geq \nu $$

Is $T : \mathbb{T}^2 \rightarrow \mathbb{T}^2$ expansive? Justify your answer.

I am struggling with this question as the usual method of see if things are expansive, ie to consider points on a vertical line then a horizonatal line doesnt work. How should I approach this question?

Answer 1

$$\forall n,\quad T^n(0,y)-T^n(0,y')=(0,y-y')\ \text{mod}\ 1\implies d(T^n(0,y),T^n(0,y'))=d((0,y),(0,y'))$$