Statement: Given $P:\Sigma\cap U\rightarrow \Sigma\cap U'$ a Poincaré map related to a T-periodic orbit $\gamma = \left \{ \varphi (t,p) : t\in [0,T] \right \}$ of a vector field $X:W=\overset{\circ}{W}\subset\mathbb{R}^n \rightarrow \mathbb{R}^n; \: C^r, \:r\geq 1$. Show that in an appropriate basis:
where $\tau (q)$ is the returning time for $q\in \Sigma\cap U$.
First approach: I see that the first column is for the eigenvalue 1 that has eigenvector X(p) because: $X(p) = X(\varphi(t,p))_{\mid t = 0} = \frac{\mathrm{d} }{\mathrm{d} t}\varphi(T,\varphi(t,p))_{\mid t = 0} = D\varphi(T,\varphi(t,p))\cdot \frac{\mathrm{d} }{\mathrm{d} t}\varphi(t,p)_{\mid t = 0}=D\varphi(T,p)\cdot X(p).$
But now, I don't know what to do with the other part of the matrix. I would appreciate any help. Thank you!
