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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

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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.

Ѕᴀᴀᴅ
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Noob mathematician
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  • Would you please define "Alekseev Cone Field" for those like me who are not familiar with it? 2. I'd suggest you make your referencing more precise. Perhaps, adding the number of the page on which this proof is written (or a link to the book) could help.
    • – Arman Malekzadeh Jun 19 '23 at 20:14
  • @ArmanMalekzadeh I have added the page number and link – Noob mathematician Jun 20 '23 at 05:53