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Dynamics in One Complex Variable. (AM-160), 3rd Edition by John Milnor

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§8. Geometrically Attracting or Repelling Fixed Points

The next four sections will study the dynamics of a holomorphic map in some small neighborhood of a fixed point. This local theory is a fundamental tool in understanding more global dynamics. It has been studied for well over a hundred years by mathematicians such as Ernst Schröder, Gabriel Kœnigs, Léopold Leau, Lucjan Böttcher, Pierre Fatou, Gaston Julia, Hubert Cremer, Carl Ludwig Siegel, Thomas Cherry, Alexander Bryuno, Jean Écalle, Serguei Voronin, Michel Herman, Jean-Christophe Yoccoz, and Ricardo Perez-Marco. In most cases it is now well understood, but a few cases still present extremely difficult problems.

We start by expressing our map in terms of a local uniformizing parameter ...

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