How to apply the Chain Rule for manifolds to $df^n_x$?

Question

Let $M$ be a smooth manifold, $f$ a $C^1$ diffeomorphism and $x \in M$. I don't understand how to apply the Chain Rule for manifolds to $df^n_x$, where $f^n = f \circ \cdots \circ f$ $n$ times. Can someone help me?

Answer 1

$df^n_x:T_xM \rightarrow T_{f^n(x)}M$

The chain rule can be aplied in the following way

$df^n_x(v) = df_{f^{n-1}(x)} \circ df_{f^{n-2}(x)} \circ df_{f^{n-3}(x)} ... \circ df_{x}(v) $

I hope this helps.

Answer 2

Well, just like in single-variable calculus, just do $d(f^{\circ n})_x = df_{f^{\circ(n-1)}(x)}\circ d(f^{\circ(n-1)})_x$. For example: $$d(f^{\circ 3})_x = df_{f^{\circ 2}(x)}\circ d(f^{\circ 2})_x = df_{f^{\circ 2}(x)} \circ df_{f(x)}\circ df_x,$$and so on.