Let $T : S^1 \times S^1 \to S^1 \times S^1$, $T(x, y) = (x + \alpha, y + x) \; mod \; 1$. For every $n \in \mathbb{N}$ $$ T^n(x, y) = (x + n \alpha, y + nx +\frac{n(n-1)}{2} \alpha) \; mod \; 1. $$ I want to calculate the topological entropy of $T$.
How can I find an $(n, \epsilon)$-spanning set?
Thank you!
Note that $$ T^n(x,y)-T^n(x',y')=(x-x',y-y'+n(x-x'))\bmod1 $$ is independent of $\alpha$. This shows that $(n,\varepsilon)$-spanning sets are independent of $\alpha$ and so the same goes for the entropy. Now note that the map $$ T_0(x, y) = (x, y + x)\bmod1 $$ is a toral automorphism having only $1$ has eigenvalue. Hence, $h(T)=h(T_0)=\log 1=0$.