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When we speak of local behavior, we mean the behavior of a complex analytic function near some point. In this set of notes, we examine the behavior of a complex analytic function near a fixed point or, by extension, near the points in a periodic orbit.

The simplest periodic orbit is that of an attractive fixed point. Note that this excludes the super-attractive case. While the behavior near a super-attractive fixed point has some qualitative similarities to the behavior near a fixed point, there are important differences and the analysis is somewhat different.

On the other hand, once we understand the behavior of an attractive fixed point, it's quite easy to extend that understanding to attractive orbits by examining the function \(F=f^n\text{.}\) We'll also be able to use our analysis of attractive fixed points to understand repelling fixed points by considering \(f^{-1}\text{.}\)

The most subtle type of fixed point is a neutral fixed point, where \(|f'(z_0)|=1\text{.}\) In this case, \(f'(z_0)=e^{i\theta}\) and there are a variety of possibilities depending on the number theoretic properties of \(\theta\text{.}\)

It's worth emphasizing that the results here apply to complex analytic functions in general - polynomials, rational functions, transcendental functions, whatever. Thus, we can use these results when exploring the iteration of a truly wide variety of functions.