I was reading the following proof of "Alekseev Cone Field Criterion" from p.225 of the book Hyperbolic Flows by Boris Hasselblatt and Todd Fisher
I was trying to verify the "Only if" part of the proof as it is said that it follows from definitions. We consider the splitting $E^c_x\oplus E^s_x\oplus E^u_x$ since $\Lambda$ is a hyperbolic set.
Consider $v(=v^c+v^s+v^u)\in C_\beta(E^u_x,E^c_x\oplus E^s_x)$ then $\|v^c+v^s\|\le \beta\|v^u\|$, and $v^c=k\dot\varphi(x)$ for some $k\in \mathbb R$
Also we know there exists $\lambda\in (0,1)$ and $C>1$ such that $\|D\varphi^t(v^s)\|\le C\lambda^t\|v^s\|$ and $\|v^u\|\le C\lambda^t\|D\varphi^{t}(v^u)\|$. Moreover $D\varphi^t(v^c)=k\dot\varphi(\varphi^t(x))$.
Then we get $D\varphi^t(v^c+v^s)=k\dot\varphi(\varphi^t(x))+D\varphi^t(v^s)$
How do I take care of this norm $\|k\dot\varphi(\varphi^t(x))+D\varphi^t(v^s)\|$?
Suppose I considered $k=0$ and proceed then I get
$$\|D\varphi^t(v^s)\|\le C\lambda^t\|v^s\|\le \beta C\lambda^t\|v^u\|< \beta \|v^u\|\le \beta\lambda^{t}\|D\varphi^{t}(v^u)\|\le \beta\|D\varphi^{t}(v^u)\|$$ for $\beta\in (0,1)$ choosen appropriately.
Please help me and let me know if I am doing anything wrong here, I am new to this topic.