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Author Topic: Autumn Forest I  (Read 822 times)
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Fractal Senior
Posts: 2592

« on: November 02, 2012, 12:20:34 PM »

Autumn Forest I


This is the start of a zoom sequence in Supernova, but with different parameters from Spring Garden. In particular, b is well away from zero, so the "lakes" inside the seahorses are not superattracting basins. They may get very gnarly or even shatter, as we shall see.

In this initial image, the brown identifies escaping points, green identifies points that go to attractors of period 3, and dark blue to black represent finite attractors of all other periods. The upper left region is a large minibrot. The lower left region is the main escaping region. The upper right region would, in a b = 0 Supernova Mandelbrot, have been a mirror of the escaping region but going to a superattractor at zero instead. With b moved away from zero, this basin bifurcates and in fact the bifurcations create a second minibrot, mostly hidden "underneath" the first (choosing zero as our critical point for this map would reveal it fully). The large green region is the intersection between one of its period-3 buds and the "zero basin" of the upper right part of the image. In the "zero basin", there is a period 3 attractor in the green region; the dark buds bordering the green region have other "zero basin" periods; and the brown beyond it has the "zero basin" fail to even properly exist. The points that would go to it just bounce around a foamy region until they hit one of the inverse "lakes" of escaping points and escape, and this dynamical-plane structure is, as usual, reflected in the parameter plane: that region is a foam of escape-lakes and a measure-zero set of Misiurewicz points and less-rational dendrite points.

The "spike" that would normally lie on the x axis has also been bent and twisted by b.

The zoom target is centered and is along the border between the escape basin and the zero basin in the middle of the period-3-zero-basin region. (The period-3 green can only easily be seen on the zero basin side of the divide, though.) The "spike" passes through here, dotted on the lower side with seahorses.

The colors are suitable for the season, are they not?

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