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Section 3.5 The Mandelbrot set

Theorem Theorem 3.4.1 expresses a fundamental dichotomy amongst quadratic Julia sets. There are two types: connected and totally disconnected. Furthermore, the type that arises from a particular value of \(c\) depends completely on the orbit of the critical point. Thus, it might be fruitful to partition the parameter plane into two portions: the portion where the critical orbit stays bounded and the portion where the critical orbit diverges to infinity. This partition leads to one of the most iconic images from the study of chaos - the Mandelbrot set.

Definition 3.5.1 The Mandelbrot set

The Mandelbrot set is the set of all complex parameters \(c\) such that the Julia set of the function \(f_c(z)=z^2+c\) is connected. Equivalently, by theorem Theorem 3.4.1, it is the set of all \(c\) values such that the orbit of zero under iteration of \(f_c\) is bounded by \(2\)

Theorem Theorem 3.4.1 yields an algorithm to generate images of the Mandelbrot set that's very similar to our algorithm for generating Julia sets of polynomials.

The results of this algorithm are shown in figure Figure 3.5.3. While the algorithm is very similar to our algorithm for generating Julia sets, it's very important to understand the distinctions. The Mandelbrot set lives in the parameter plane, while Julia sets live in the dynamical plane. For the Julia set of a quadratic function \(f_c\text{,}\) we fix the value of \(c\) and explore the behavior of the orbit of \(z_0\) for a grid of choices of \(z_0\text{.}\) For the Mandelbrot set, we explore the global behavior of \(f_c\) by examining the critical orbit; we do this for a grid of choices of \(c\text{.}\)

Figure 3.5.3 The Mandelbrot set