On the definition of Anosov diffeomorphism

Question

I have a question about the definition of Anosov diffeomorphism. It might sound a little bit silly, but anyway.

In most definitions I saw, we say that $f: M \to M$ is an Anosov diffeomorphism if there exists a decomposition $T_x M = E^s(x) \bigoplus E^u(x)$, a constant $C > 0$ and $0< \lambda < 1$ such that $\forall v \in E^s(x), \forall n \geq 0 : \Vert d_xf^n v \Vert \leq C \lambda^{n} \Vert v \Vert$ ... My question is : by $ d_xf^n v$, do we mean $d_x(f^n)$ (that is, the differiential of $f^n$ at $x$ , or $(d_xf)^n$, that is the differential of $f$ at $x$ composed with itself $n$-times. What difference does it make?

Thanks a lot !

Answer 1

$df_x:T_xM\rightarrow T_{f(x)}M$, so you cannot compose $(df_x)^n$ unless $f(x)=x$.