Group action a Compact Hausdorff space

Question

Let $X$ be a compact Hausdorff topological space and $G$ be a group acting on $X$ continuously. Is it true that orbits $Gx$ are compact subsets of $X$? In fact my action is like this. Let $\phi$ be a homeomorphism of $X$ and the group $\mathbb{Z}$ on $X$ via $\phi$.

Answer 1

Not necessarily. Consider the action of $(\Bbb R,+)$ on $S^1\times S^1$ defined by $t.(z_1,z_2)=(e^{it}z_1,e^{it}z_2)$. If $x(e^{\alpha i},e^{\beta i})$ with $\alpha,\beta\in\Bbb R$, $\alpha\neq0$, and $\frac\beta\alpha\notin\Bbb Q$, then the orbit of $x$ is dense but it is not the whole $S^1\times S^1$; in particular, it is not compact.