Complex dynamics
Multibrot sets
Multibrot sets are the parameter spaces for the polynomial family fc(z) = zn + c. The classic Mandelbrot set is the case n = 2. For higher integer exponents the set develops n − 1-fold rotational symmetry.
At a glance
| Iteration | zn+1 = znd + c, z0 = 0 |
|---|---|
| Rotational symmetry | (d − 1)-fold |
| Connectivity | Connected for all d ≥ 2 (Branner-Hubbard) |
| Large-d limit | Approaches the closed unit disk |
Symmetry and structure
Replacing c → e2πi/(d-1) c commutes with the iteration in a way that produces (d − 1)-fold symmetry of the parameter set. For d = 2 this is trivial (1-fold; only reflection in the real axis); for d = 3 the set has three-fold symmetry, and so on.
Connectivity
Branner and Hubbard generalized Douady-Hubbard’s connectivity proof: every Multibrot set is connected. They are not, however, always locally connected; this remains an open question for general d.
References
- Branner, B. & Hubbard, J. H., “The iteration of cubic polynomials I,” Acta Math., 1988.
- Douady, A. & Hubbard, J. H., “Étude dynamique des polynômes complexes,” Publ. Math. d’Orsay, 1984/1985.
- Mandelbrot set
Try it
Run an interactive playground at /tools/multibrot.