The Spider Algorithm

One of the reasons complex analytic dynamics has been such a successful subject is the deep relation that has surfaced between conformal mapping, dynamics and combinatorics. The object of the spider algorithm is to construct polynomials with assigned combinatorics. This shows up when you try to understand the Mandelbrot set. For this discussion we will write our quadratic polynomials Qc(z) = z + c. Every such polynomial has a filled in Julia set Kc, formed of the points with bounded orbits under iteration of Qc. A result of Fatou asserts that if the critical point 0 ∈ Kc, then Kc is connected, and if 0 / ∈ Kc, then Kc is a Cantor set. By definition, the Mandelbrot set M is the set of c for which Kc is connected. Let D denote the open unit disc, and let ΦM : C−M → C−D be the conformal mapping which maps ∞ to ∞ and is tangent to the identity at infinity. The existence of this mapping is not obvious; it is proved to exist at the same time as the