Let $M$ be a smooth manifold, $f:M\to M$ a diffeomorphism. An $f$-invariant set $\Lambda\subset M$ is called hyperbolic if there are constants $c>0$ and $0<\lambda<1$ such that: $$T_pM=E_p^s\oplus E_p^u\text{ for all }p\in \Lambda;$$ $$Df_p(E^s_p)=E_{f(p)}^s\text{ and }Df_p(E^u_p)=E_{f(p)}^u\text{ for all }p\in\Lambda$$ $$||Df_p^n(v)||\leq c\lambda^n||v||\text{ for all }v\in E_p^s,n\in\mathbb{N}$$ $$|||Df_p^{-n}(v)||\leq c\lambda^n||v||\text{ for all }v\in E_p^u,n\in\mathbb{N}$$
I've taken this definition from this article.
First question: the article says $Df_p$ must be an "expansion" on $E^u_p$, but I'm not sure what that means. Assuming the definition above and using the relation $f^{-1}\circ f=id$ and the chain rule, I've checked that $||Df_p(v)||\geq \frac{1}{c\lambda}||v||$. Is that what he means by expansion? I think this would only make sense for $\frac{1}{c\lambda}>1$, but that's not necessarily true.
Second question: why must we say $||Df_p^n(v)||\leq c\lambda^n||v||$ for all $n$? Wouldn't it be enought to say it for $n=1$? I mean, since $Df_p^n=Df_{f^{n-1}(p)}\circ...\circ Df_p$, the general statement would follow by induction, right?