Let me define by $R_\alpha: S^1\rightarrow S^1$ to be the rotation on the circle by the angle $\alpha$. Now consider $\alpha, \beta$ and $R_\alpha, R_\beta$ and define $$M:S^1\times S^1\rightarrow S^1\times S^1;~~(x,y)\mapsto (R_\alpha x, R_\beta y)$$ Assume that $\alpha=2\beta$ I need to show that $O:=\{M^n(x,y):n\geq 0\}$ is not dense in $S^1\times S^1$.
I first remark that $$M^n(x,y)=(R_\alpha^n x, R_\beta^n y)=(R_{2\beta}^n x, R_\beta^n y)=(R_\alpha^{2n} x, R_\beta^n y)$$hence $$O=\{(R_\alpha^{2n} x, R_\beta^n y):n\geq 0\}$$Now I know that I need to find a point $(u,v)\in S^1\times S^1$ such that there exists no sequence $(R_\alpha^{2n_i} u, R_\beta^{n_i} v)\rightarrow (u,v)$, but I don't see how to find such a $(u,v)$ can someone give me a hint?
Let $a:=e^{i\alpha}$ and $b:=e^{i\beta}$. $$M(x,y)=(ax,by)=(b^2x,by)$$So: $$M^n(x,y)=(b^{2n}x,b^ny)$$
Let's say we have an arbitrary pair $(x,y)$. We want to show that the set: $$\{(b^{2n}x,b^ny):n\in\Bbb N\}$$Is not dense in the torus.
Because multiplication is a homeomorphism, I just want to show that $\{(b^{2n},b^n\gamma)\}$ is not dense where $\gamma:=yx^{-1}$. I advise you to consider $(u,v)=(1,i\gamma)$.