I am trying to solve the following problem. Take the torus $\mathbb{T}^{2}$ and define the map $T(x,y)=(x + \alpha$ mod 1, $x+y$ mod $1)$, where $(x,y) \in [0,1]^{2}$.
By induction, we have $T^{n}(x,y) = (x + n\alpha$ mod 1, $y+nx +\frac{n(n-1)\alpha}{2}$ mod $1)$ for $n \in \mathbb{N}$. I would like to show that:
a) for integer $k$ such that $\frac{1}{k} < \frac{\epsilon}{3}$, the set $$S = \{(\frac{i}{nk},\frac{j}{k}), 0 \leq i < nk, 0 \leq j <k\}$$ is an $(n,\epsilon)$-spanning set
b) find the topological entropy of $T$
c) find if $T$ is expansive or not.
For a) if $\mathbf{x}=(x,y) \in [0,1]^{2}$, then there is $i$ and $j$ such that $$x \in [\frac{i}{nk}, \frac{i+1}{nk}] \times [\frac{j}{k}, \frac{j+1}{k}]$$.
So if $\mathbf{y} \in [0,1]^{2} \in S$ is at the end points of the above rectangle, we have for $k$-th iterate of $T$,
$d(T^{k}(\mathbf{x},T^{k}(\mathbf{y})) \leq \sqrt{\frac{1+n^{2}}{n^{2}k^{2}} + \frac{k^{2}}{n^{2}k^{2}} + \frac{2k}{nk^{2}}} \leq \frac{\sqrt{4n^{2}+1}}{nk} < \frac{\sqrt{5n^{2}}}{nk} < \frac{3}{k} <\epsilon$, which is what we require for the definition of a spanning set.
b) and since $h_{top}(T) = \lim_{\epsilon \longrightarrow 0} \lim_{n \longrightarrow \infty}sup \frac {\log (Span(n,\epsilon)}{n}$ we have Span $(n, \epsilon)=k$ and thus $h_{top}(T)=0$. (Here $h_{top}$ stands for topological entropy).
c) $T$ is NOT expansive. Take two points with the same $x$ coordinate, i.e. $(x_{1},y_{1})$ and $(x_{1},y_{2})$ and then we have $d(T^{n}(x_{1},y_{1}), T^{n}(x_{1},y_{2})) = $ min { $|y_{1}-y_{2}|, 1-|y_{1}-y_{2}|\} <\epsilon$ if we take $|y_{1}-y_{2}|<\epsilon$.
Are my arguments above correct? If not, I'd be grateful if you could point out the errors and provide the missing details.