Timeline for Who proved that the Mandelbrot set's Julia sets are locally connected?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Dec 28, 2019 at 0:34 | comment | added | Lasse Rempe | @AlgebraicsAnonymous As you might expect, the set of parameters where the Julia set is locally connected is dense in the boundary, as is the set of parameters where it is not locally connected. Indeed, every boundary of every hyperbolic component contains points of both types, and it is well-known that little Mandelbrot copies accumulate at every point of the boundary. [This is overkill, as much less is required, but it may give the conceptually clearest picture.] | |
| Aug 29, 2019 at 22:12 | comment | added | Jacques Carette | Oops, 'interior' should have been 'boundary', fixed now, thanks. | |
| Aug 29, 2019 at 22:11 | history | edited | Jacques Carette | CC BY-SA 4.0 |
fix point 3 to say 'boundary' rather than 'interior'.
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| Aug 28, 2019 at 21:13 | comment | added | Robert Furber | @JacquesCarette The first sentences of 2 and 3 are exactly the same. Is this supposed to be this way? | |
| Aug 21, 2019 at 22:20 | comment | added | Cloudscape | If only they'd answer them... | |
| Aug 21, 2019 at 12:38 | comment | added | Jacques Carette | See the other answer for answers to your comments. I'm not sure about the status of your last question, I last actively worked in the area more than 20 years ago and didn't keep up with all results. | |
| Aug 20, 2019 at 6:27 | comment | added | Cloudscape | Follow-up question: Is the set of those boundary parameters whose Julia set is locally connected dense in the boundary? | |
| Aug 20, 2019 at 6:10 | comment | added | Cloudscape | By the theorems in the Orsay notes, this would mean that there are parameters in the boundary of M for which there is no indifferent or attracting cycle to which the orbit of zero converges, and neither a repellent cycle in which it ends up after a finite amount of time. | |
| Aug 20, 2019 at 6:00 | comment | added | Cloudscape | So are you saying that some Julia sets that correspond to the boundary of the Mandelbrot set are not locally connected? (I think there might have been copy&paste issues.) | |
| Aug 19, 2019 at 21:26 | history | answered | Jacques Carette | CC BY-SA 4.0 |