I'm reading the book Differential Dynamical Systems and the last statement of the following proof confuses me a lot: Proof of Poincaré's map theorem
I don't see why $f(x_0)$ is the eigenvector of the monodromy matrix with eigenvalue one. Could anyone explain a bit?
Let $x$ be a point on $\gamma$. Since $\gamma$ is periodic, $x$ has the same period as $x_0$, and in particular $\varphi_{\tau(x_0)} (x) = x$. But the points on $\gamma$ are exactly $\{\gamma_t (x_0) : t \in \mathbb{R}\}$. Hence, for all $t \in \mathbb{R}$,
$$\varphi_{\tau(x_0)} (\varphi_t (x_0)) = (\varphi_t (x_0)).$$
Now, differentiate this identity at $t=0$, to get:
$$D \varphi_{\tau(x_0)} (x_0) \cdot \varphi_t' (x_0) = \varphi_t' (x_0).$$
But, by definition of the flow, $\varphi_t' (x_0) = f(x_0)$, and by definition of the monodromy matrix, $M=D \varphi_{\tau(x_0)} (x_0)$. Hence,
$$M(f(x_0)) = f(x_0).$$
The basic idea is that points along the same orbit stay roughly at the same distance on from another (as long as the space is compact, the flow regular enough with no fixed point...), so the derivative in the direction of the flow will be close to $1$. This can easily be made exact for periodic orbits.