What is given:
Let K be a irrational number and $f:T^2 \rightarrow T^2$ be the homeomorphism of the 2-torus given by $f(x,y)=(x+K,x+y)$.
The exercise:
Show that for every non-empty, open, $f$-invariant set is dense.
What I did:
I did note that I need to prove that $f$ is topological transitive. So I need to find a point in $T^2$ whose forward orbit is dense in $T^2$.
First I picked the point $(a,b)\in T^2$ with $a,b \in \mathbb{Q}$. Then I find that $f(a,b)=(a+K,a+b)$. I do note that the first ordinate is a irrational number and the second one a rational number. Now with the pigeon-hole principle we find that for any $\epsilon >0$ that there are $m,n< \frac{1}{\epsilon}$ such that $m<n$ and $d((x+K)^m,(x+K)^n)<\epsilon$, where $(x+K)^m=(x+k) \circ ...\circ (x+K)$, m times. Now we have that $(x+K)^{n-m}$ is a rotation by an angle less than $\epsilon$. Thus we find that every forward orbit is dense in $\mathbb{R} / \mathbb{Q}$.
Now we find that for $(x,y)\in T^2$ its forward orbit is dense in $T^2$. Thus $f$ is topological transitive.
My question to you: I do not belive that my prove it 100% correct. So I was wondering if you guys could give my some tips and maybe could correct me.