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Section 3.6 The components of the Mandelbrot set

By definition, the Mandelbrot set decomposes the complex parameter plane into two components corresponding to a very coarse decomposition of the types of behavior of \(f_c\text{.}\) A glimpse at figure Figure 3.5.3 (or any of the many images of it available on the web) shows that it consists of rather conspicuous subparts. It appears that there is a large “main cardioid” and that, attached to that main cardioid are a slew of disks or near disks. If we zoom in, we see many smaller copies of the whole set. As it turns out, all these components correspond to particular dynamical behavior. Values of \(c\) chosen from within one component yields functions \(f_c\) with similar qualitative behavior. As \(c\) moves from one component to another, that qualitative behavior changes; we say that a bifurcation occurs. In this section, we'll describe the most conspicuous behavior that we see.