Mandelbrotins

Mandelbrotins are a family of fractal sets in the complex plane that generalize the classical Mandelbrot set. For each integer degree d ≥ 2, define the polynomial-like map f_c(z) = z^d + c with z_0 = 0. The Mandelbrotin of degree d, denoted M_d, is the set of complex parameters c for which the orbit {z_n} remains bounded under iteration z_{n+1} = z_n^d + c. The classical Mandelbrot set corresponds to d = 2.

Geometrically, M_d resembles the Mandelbrot set but with distinct combinatorial structure that depends on d. For small d, the cardioid-like main body persists with bulb-like attachments; for larger d, the parameter plane develops more sectors and cusp-like features. The boundary of M_d is fractal, and the interior consists of hyperbolic components where the orbit tends to attracting cycles. M_d is connected for all d, mirroring the Mandelbrot property.

Computationally, Mandelbrotins are rendered using escape-time or distance-estimate methods. Iteration is performed for a grid of c-values, and the iteration sequence is terminated when |z_n| exceeds a fixed bound or when a maximum number of iterations is reached. Coloring by escape time highlights the boundary complexity and internal structure.

Extensions include higher-dimensional generalizations using multivariable or quaternionic polynomials, and the associated Julia sets J_c^(d). Mandelbrotins are popular in fractal art and dynamical systems studies as a family that illustrates how changing the degree alters global dynamics while preserving core features of the original Mandelbrot set.

See also: Mandelbrot set, Julia set, fractal, complex dynamics.