Wheels Within Wheels  
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Description: The dense nesting of rings of shapes around a deep mini (10^{300}).
Stats: Views: 21 Total Favorities: 0 View Who Favorited Filesize: 934.28kB Height: 960 Width: 1280 Posted by: pauldelbrot February 28, 2019, 02:28:25 AM Rating: by 1 members. Total Likes: 1 Image Linking Codes


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marcm200  February 28, 2019, 01:36:40 PM Nice! A (somewhat unrelated question): Is there a limit of how close those minis can get to each other (compared to their size)? 
pauldelbrot  February 28, 2019, 04:37:21 PM There's no limit to how close two minis can be. There'll be a mini within any epsilon of the one visible above ... but it will be far tinier, order of epsilon squared smaller, I think. For closetogether minis that are close to the same size (so you can see the shapes of both at once in a single image of nonastronomical resolution) you need to look shallow, at the brambles you find by zooming only a little bit at the edge of the set. Or equivalent brambles attached to a mini, such as the one above. Look at the "tentacles" coming off the sides of the big bulb, or the "starfish" north and south of the seahorse valleys, and you'll find views with many recognizable minis at once. If you go beyond zedsquared you can get closer groupings. One way is the cubic and higher powers. With powers above 10 it's hard to find a view of the edge that doesn't have many recognizable minis, even after quite a ways of zooming. Another is with functions that have multiple critical points, such as matchmaker and triple matchmaker. If images using two or more of the critical points and the same zoom rectangle are superimposed minis "belonging" to distinct critical points can be found very close together and even overlapping. Parameter points from inside the overlap region will produce Julia fractals bounding the basins of more than one finite attractor. 
claude  February 28, 2019, 05:36:20 PM There may be a way to quantify the size decrease of minibrots in sequences approaching Misiurewicz points by using the multiplier of the point (derivative of periodic part of the cycle). I guess the size of minibrots decreases by m^2 as the decorating filaments decrease by m. I believe this has been experimentally verified (and perhaps even proven mathematically?) for the Misiurewicz point 2, with multiplier 4, so each successive minibrot toward the antenna is 4 times nearer the tip and 16 times smaller, 