For Christmas, I was handing out fractal Christmas Cards (they had images made in gnofract4d with the equation beneath… I gave my mom a handbound book with pictures of fractals made in sage in it since I didn’t get the latest revision of the math textbook done in time to get that). Thus, I made many fractals. At the end, I noticed something.
First I’ll make a (IMHO, reasonable) conjecture:
Now, it’s often interesting to look at julia sets when they fracture, thus they make good pictures. And it is easy to find the point of fracture on the positive real number line.
The orbits of these points exhibit a rather simple behaviour: converges to a value that satisfies if it is possible.
So we want to find the maximum value of (ie. where is the supreumum).
This is trivial to find: and if we plug that into we get