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Chapter 3 The complex quadratic family

In this chapter, we turn to complex dynamics and focus our attention on the complex quadratic family:

\begin{equation*} f_c(z)=z^2+c. \end{equation*}

In spite of its simplicity, much of the full scope of complex dynamics and chaos arises in this family. Furthermore, this is exactly the context in which the Mandelbrot set arises.

Note that \(\{f_c\}\) forms a parametrized family of quadratics that we discussed in some detail in the context of real iteration in section Section 2.6. Now, though, the variable \(z\) and the parameter \(c\) are complex valued.

We will be studying complex dynamics freely throughout the rest of the text so, if you're not on top of complex variables, now would be a good time to study up on it. One does not need to fully digest the complete body of knowledge of complex variables in order to follow the topics in this text. One just needs a basic understanding of the algebra and geometry of the complex plane, together with an understanding of how polynomial functions work. I recommend studying chapters 1 and 2 of A First Course in Complex Analysis by Mathias Beck et. al. When moving beyond the dynamics of polynomials, some information from chapter 3 would also be useful - particular, section 3.1 on Möbius transformations.