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What is the conjectured topology of the Mandelbrot set if MLC is true? My understanding is that there's a certain number of bulbs, each centred around a point
by OscarCunningham 3y ago
What is the conjectured topology of the Mandelbrot set if MLC is true?
My understanding is that there's a certain number of bulbs, each centred around a point which becomes periodic with period p after k steps. But how do they all stick together?
- bongodongobob 3y agoThe entire set is connected iirc.
- qazxcvbnm 3y agoBut what would be its homology, for instance?
- OscarCunningham 3y agoIt's known that it's connected and simply connected. So if it's locally connected then I think it has to be contractable.
- clintonc 3y agoMLC stands for "Mandelbrot Locally Connected". It's not obvious, but this is equivalent to the bulbs of the Mandelbrot set (the domains of parameters where almost all points get attracted toward periodic orbits) are dense in the Mandelbrot set. Everyone believes it to be true.
- OscarCunningham 3y agoYes, but how exactly are the bulbs arranged? Wikipedia says 'Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. Such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy'. Which sequences of bulbs have little copies at the end of them? And how do the little copies attach?
- clintonc 3y agoThe combinatorics of how the Mandelbrot set is put together is well-studied, and rather independent of MLC. The arrangement of the bulbs on the boundary of the "main cardiod" (which is where there is an attracting fixed point) is described here: https://en.wikipedia.org/wiki/Mandelbrot_set#Main_cardioid_and_period_bulbs https://en.wikipedia.org/wiki/Mandelbrot_set#Main_cardioid_a.... Generally, the patterns are given by something called Lavaur's Algorithm; see https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/lavaurs https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_com... for some explanation. Attachment points are always at the "root" of the Mandelbrot set, which is the cusp of the main cardioid. A consequence of MLC is that the combinatorial picture given by Lavaur's algorithm and related analyses is "complete" -- all dynamical information is available from the combinatorial models.
- OscarCunningham 3y agoThank you, that's very helpful!