Periodic Orbits of Arbitrarily Small Period for a Flow Without Fixed Points

Suppose for contradiction that for each $\epsilon > 0$, there exists a time $t \in (0, \epsilon)$ and a point $p \in M$ for which $\varphi_t(p) = p$. This implies that, for each $n \in \mathbb N$, there exists a time $t_n \in (0, \frac{1}{n})$ and a point $p_n \in M$ for which $\varphi_{t_n}(p_n) = p_n$. Since $M$ is compact, $\{p_n\}$ has a convergent subsequence $\{p_{n_k}\}$ with limit $p \in M$.

Now, fix a time $t > 0$. There exists a sequence of integers $\{a_n\} \subseteq \mathbb Z$ for which $\{a_n t_n\}$ converges to $t$ as $n \to \infty$. To see this, consider the sequence $\{a_n\} = \lfloor{\tfrac{t}{t_n}}\rfloor$, which is well-defined since $t_n \neq 0$ for each $n$. One always has $| \lfloor{\tfrac{t}{t_n}}\rfloor - \tfrac{t}{t_n}| < 1$. Now, consider: \begin{equation} |a_n t_n - t| = | \lfloor{\tfrac{t}{t_n}}\rfloor t_n - t| = |t_n| \cdot | \lfloor{\tfrac{t}{t_n}}\rfloor - \tfrac{t}{t_n}| < |t_n|. \end{equation} Passing to the limit, we conclude $a_n t_n \to t$ as $n \to \infty$. For each $n \in \mathbb N$, $\varphi_{t_n}(p_n) = p_n$ implies $\varphi_{a_n t_n}(p_n) = p_n$, since $a_n \in \mathbb Z$. Thus, we conclude that both \begin{equation} \lim_{k \to \infty} \varphi_{a_{n_k} t_{n_k}}(p_{n_k}) = \varphi_t(p) \text{ and } \lim_{k \to \infty} \varphi_{a_{n_k} t_{n_k}}(p_{n_k}) = \lim_{k \to \infty} p_{n_k} = p. \end{equation} Thus, $\varphi_t(p) = p$. Since $t > 0$ was arbitrary, this implies that $p$ is a fixed point of $\varphi_t$, a contradiction.