Milnor'sLemma - crowlogic/arb4j GitHub Wiki

Milnor's Lemma is a result in the field of complex dynamics, which studies the behavior of functions on the complex plane. The lemma, named after John Milnor, a renowned American mathematician, relates the concepts of simple roots and superattractive fixed points for holomorphic (complex differentiable) functions.

The lemma can be stated as follows:

Let $f(z)$ be a holomorphic function defined in a neighborhood of a point $z_0$ in the complex plane, with a simple root at $z_0$, i.e., $f(z_0) = 0$ and $f'(z_0) \neq 0$. Then $z_0$ is a superattractive fixed point of the function $g(z) = z - \frac{f(z)}{f'(z)}$.

A superattractive fixed point is a fixed point that attracts nearby points at an exponential rate. In other words, if $z_0$ is a superattractive fixed point of $g(z)$, then there exists an integer $n > 1$ such that $(g^n)'(z_0) = 0$, where $g^n$ represents the $n$-th iterate of the function $g$.

Milnor's Lemma is important in the study of complex dynamics, as it connects the concepts of simple roots and superattractive fixed points, and is frequently used in the analysis of iterative methods for finding roots, such as Newton's method.