Compact space, continuous dynamical system, stationary point I'm having trouble proving that if $X$ is a compact metric space and every continuous function $f : X \rightarrow X$ has a fixed point, then every continuous dynamical system $ \varphi $ on $X$ has a stationary point  - there exists an $x_0 \in X$ s.t. $\varphi(t, x_0) = x_0$ for every $t \in \mathbb{R}$.
$\varphi : \mathbb{R} \times X \rightarrow X$ - continuous s.t. 
$\varphi (0,x) = x$ for every $x \in X$
$\varphi (t, \varphi(s,x) ) = \varphi(s+t, x)$ for all $s, t \in \mathbb{R}, \ \ x \in X $
Could you help me with that? 
 A: Write $\phi_t(x) = \phi(t, x)$.  All we need is the following simple fact:
"If $\phi_t (x) = x$, then $\phi_{mt}(x) = x$ for all $m\in \mathbb N$"
This fact follows from the second property of $\phi$. In particular, we have 
$$(*)\ \{x :\ \phi_{t/m}(x)=x\} \subset \{x :\ \phi_t(x)=x\} $$
Now we proceed to show the assertion. By continuity of $\phi$, it suffices to show that there is $x\in X$ such that $\phi_q(x) = x$ for any $q\in \mathbb Q^+$. Let $q = n/m$, by (*) (by putting $t=m/n$), it suffices to find $x\in X$ such that $\phi_{1/n}(x) = x$ for all $n\in \mathbb N$.
Let $\{p_1, p_2, \cdots\}= \{2, 3, \cdots\}$ be the enumeration of primes. For $l \in \mathbb N$, let 
$$B_l = \{x:\ \phi_{(p_1^{k_1}\cdots p_l^{k_l})^{-1}} (x) = x\ \ \text{for all }k_1, \cdots k_l \in \mathbb N\}\ .$$
Then $B_{l+1} \subset B_l$ by (*). If we can show that each $B_l$ is nonempty, then as $X$ is compact and $B_l$ are all closed, $\bigcap_l B_l$ is nonempty and we are done.
Claim: $B_l$ is nonempty. 
Proof: Let 
$$A(k_1, \cdots, k_l) = \{x: \phi_{(p_1^{k_1}\cdots p_l^{k_l})^{-1}} (x) = x\}\ ,$$
then 
$$B_l = \bigcap_{k_1, \cdots, k_l \in \mathbb N} A(k_1, \cdots, k_l)$$
Let $(k_1^i, \cdots, k_l^i)$, $i=1, \cdots m$. Then 
$$ A(\tilde k_1, \cdots \tilde k_l)\subset\bigcap_{i=1}^m A(k_1^i, \cdots, k_l^i), $$
where $\tilde k_j = \max\{k_j^1, \cdots, k_j^m\}$ for $j=1, \cdots , l$. This shows that the collection of closed sets
$$\{ A(k_1, \cdots, k_l):\ k_1, \cdots, k_l \in \mathbb N\}$$
has finite intersection property. Thus $B_l$ is nonempty as $X$ is compact. 
