Koenigs linearization theorem

Remarks

In the attracting case $0 < \abs{\lambda} < 1$, the map is given by $\varphi(z) = \lim_{n \to \infty} f^n(z) / \lambda^n$. The limit converges uniformly near $0$, because $f^n(z) = \lambda^n z (1 + O(\abs{z}))$ with errors that shrink geometrically. The repelling case $\abs{\lambda} > 1$ follows by applying the attracting case to the local inverse of $f$. So near a fixed point that is attracting but not superattracting, or repelling, $f$ is locally conjugate to its linear part $z \mapsto \lambda z$. Example: $f(z) = z / (2 - z)$ has $f'(0) = 1/2$, and $\varphi(z) = z / (1 - z)$ satisfies $\varphi(f(z)) = \varphi(z) / 2$. The hypotheses are needed. For $\lambda = 0$, the map $z \mapsto z^2$ is not conjugate to $z \mapsto 0$, since it is not constant. For $\lambda = 1$, the map $z \mapsto z + z^2$ is not conjugate to the identity, since it is not the identity.

Going deeper

For a student

The function $\varphi$ is called the Koenigs function of $f$ at the fixed point. In the attracting case it extends to the whole basin of attraction, where it still satisfies $\varphi \circ f = \lambda \varphi$. The cases left out lead to other theorems: Böttcher's theorem for $\lambda = 0$, the Leau–Fatou flower theorem for roots of unity, and Siegel and Brjuno's small divisor results for other $\abs{\lambda} = 1$.