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Parameters
Parameter plane (click to choose c)
Dynamics of polynomials in one complex variable, drawn on the GPU. Drag to pan, scroll to zoom, click to follow an orbit.
For a polynomial $f$, the filled Julia set $K(f)$ is the set of points whose orbits stay bounded, and the Julia set $J(f)=\partial K(f)$ is its boundary. It is also the closure of the repelling periodic points, and the set where the iterates $f^{\circ n}$ fail to form a normal family. For $f(z)=z^2+c$ a theorem of Fatou and Julia gives the dichotomy: $J$ is connected if the critical point $0$ has a bounded orbit, and $J$ is a Cantor set (compact, totally disconnected, perfect) if it escapes to infinity.
Three renderings are offered. Escape time colours each point by a smoothed count of the iterations needed to leave a large disk; the colour bands are level curves of the Green's function of $K(f)$, and $J$ is where the bands pile up. Distance estimate draws only points within a few pixels of $J$, using the bound $\operatorname{dist}(z,J)\lesssim |z_n|\log|z_n| / |z_n'|$; this is the right way to see a Cantor set, which has empty interior and zero area. Filled paints $K(f)$ black.
Rendering uses single-precision arithmetic in a WebGL fragment shader, so very deep zooms will eventually pixelate. The “Cantor” and “connected” labels come from iterating the critical orbit in double precision for a few thousand steps; for parameters extremely close to the boundary of the Mandelbrot set the label can be wrong in either direction, as the example $c=-0.8+0.156i$ is meant to show.
The example $f(z)=z^2-6$ is Problem 4-e in John Milnor's Dynamics in One Complex Variable; see also Devaney's An Introduction to Chaotic Dynamical Systems for the real-line Cantor construction, and Branner and Hubbard's papers on cubic polynomials for the case where exactly one critical point escapes.