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Spacefilling Seahorse Tails



http://www.fractalforums.com/index.php?action=gallery;sa=view;id=16345

A zoom into the upper left of "Spacefilling Seahorse Valley", a Triple Matchmaker Mandelbrot image. Most of the image is packed with detail, aside from the interiors of a few small minibrots.

Check out the seahorse tail at lower left. The double spirals at the left edge of the spiral are radially oriented, but follow the curve of the spiral a ways and they turn gradually to become circumferentially oriented -- just as in a standard Mandelbrot seahorse. The same basic laws of Mandelbrot doubling-zones clearly apply around the minibrots even here, and apply even to the features revealed in the seemingly-chaotic regions of parameters with whole-Riemann-Sphere Julia sets. Those features have a mathematical reality independent of "escape times", distance estimates, or many other things; they represent some underlying feature of the dynamics that exists even in the absence of any stable attractors, something that's confined to the narrow channels of dendrites in the normal Mandelbrot set but here is allowed to spread out two-dimensionally.

Something, perhaps, that the normal Mandelbrot set approximates ever more closely when zoomed ever deeper, based on how the boundary's Hausdorff dimension is 2...