Let $X=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{0\}$ be subspace of $\mathbb{R}$ with usual metric and let $(T, X)$ be a semiflow on $X$, this means that $\varphi:T\times X\to X$ with $\varphi(t, x)= tx$ is a continuous semigroup action. Also, we assume that $p\in X$ does have dense orbit, this means that $\overline{Tp}=X$.
Is it may be happen that there is $t_0\in T$, $t_0\neq e$, such that $t_0p=p$?
In the case of $\mathbb{Z}^+$, it is impossible. What can say in other case of semigroups?