I'm studying Dynamical Systems using the book "Geometric Theory of Dynamic Systems - J. Palis and W. de Melo".
On page 40, the author proposes to demonstrate the following theorem.
Theorem (Tubular Flow) Let $M \subset \mathbb{R}^n$ be a $\mathcal{C}^{\infty}$-manifold of dimention $m$, $X$ $\in$ $\mathcal{X}^r(M) = \{F: M \rightarrow \mathbb{R}^n$; $F(x)$ $\in$ $T_x M$ , $\forall x$ $\in$ $M$ and $F$ is a $\mathcal{C}^r$ function $\}$, and $p$ $\in$ $M$ be a regular point of $X$ (i.e. $X(p) \neq 0$). Let $C = \{(x^1 , ..., x^m) \in \mathbb{R}^m; |x^i|<1 $ $\}$ and let $X_C$ be the vector field on $C$ definied by $X_c (x) = (1,0,...,0)$. Then there exists a $\mathcal{C}^r$ diffeomorphism $h: V_p \rightarrow C$, for some neighbourhood $V_p$ of $p$ in $M$, taking trajectories of $X$ to trajectories of $Xc$.
Proof:
My Doubt: Can someone explain to me why the function $h$ defined on the line underlined in red, take trajectories of $X$ to trajectories of $Xc$?
I think that this is not true because $h^{-1} (t+x_1,x_2,...,x_m) = x^{-1} \circ \tilde{\psi} \circ f (t +x_1, ... , x_m)$ is not a trajectory of $X$. In fact \begin{align*} \frac{d}{dt} h^{-1} (t + x_1 ,x_2 , ... ,x_m) &= \frac{d}{dt} \left(x^{-1} (\tilde{\psi} (f (t + x_1, x_2 ,...,x_m) ) \right)\\ &= D x^{-1}_{\tilde{\psi} (\varepsilon (t + x_1, x_2 ,...,x_m))} \cdot D \tilde{\psi}_{\varepsilon (t+ x_1, x_2 , ...,x_m)} \cdot \frac{d}{dt} f(t+x_1,x_2,...,x_m) \\ &= D x^{-1}_{\tilde{\psi} (\varepsilon (t + x_1, x_2 ,...,x_m))} \cdot D \tilde{\psi}_{\varepsilon (t+ x_1, x_2 , ...,x_m)} \cdot (\varepsilon e_1) \\ &= \varepsilon D x^{-1}_{\tilde{\psi} (\varepsilon (t + x_1, x_2 ,...,x_m))} \cdot \frac{d\tilde \psi}{dt} ( \varepsilon (t + x_1,x_2,...,x_m)\\ & = \varepsilon D x^{-1}_{\tilde{\psi} (\varepsilon (t + x_1, x_2 ,...,x_m))} \cdot x_* X(\tilde{\psi} ( \varepsilon(t+x_1,x_2,...,x_m)))\\ &= \varepsilon X(x^{-1} \circ \tilde{\psi} \circ f (t + x_1,x_2,...,x_m))\\ &= \varepsilon X( h^{-1} (t+x_1,x_2,...,x_m)) \neq X( h^{-1} (t+x_1,x_2,...,x_m)),\\ \end{align*} where $x_* X(p) = Dx_{x^{-1} (p)} X(x^{-1}(p)).$
Keeping in mind the above calculation, why $h$ satisfies the conditions in the theorem?