Consider the matrix
$\pmatrix{a & b \\ c & d} ^n$
This is isomorphic to the $n$ th iteration of the Möbius transform $\frac{a z + b}{c z + d}$ when the determinant is nonzero.
So I wonder what is the analogue isomorphism from a power of a $3 \times 3$ matrix to the iteration of an analytic function ?
I prefer to stay on the complex plane, so I am not so intrested in iterating functions defined for noncomplex numbers or more than 2 dimensions.