I have a (sufficiently smooth) function $f$ that is volume-preserving (Jacobian has determinant $1$) and invertible. Given two integers $\ell, k \in \{0, 1, \ldots, T\}$ with $\ell > k$ it seems possible to say $f^\ell(x)$ can be sufficiently similar to$f^k(x)$ when $\ell$ and $k$ are not too dissimilar.
I would like to write $f^\ell(x)$ in terms of $f^k(x)$.
I tried something like a Taylor expansion, but I think I am Taylor expanding the wrong thing. I am happy to assume extra conditions if necessary. $$ f^{\ell - k}(x) \approx f^{\ell - k}(f^k(x)) + (x - f^k(x))\frac{d}{dx}f(f^k(x)) + (x - f^k(x))^2 \frac{d^2}{dx^2} f(f^k(x)) + \cdots $$
The problem with this is that it's actually $f^{\ell - k}$ and not $f^\ell$.