On 06/09/2020 11:20, I wrote:
On 06/09/2020 01:07, Allan Wechsler wrote:
Note that the set of Foias points looks much more like a Julia set than it does like a Mandelbrot set, which makes sense because it is asking about the divergence of a *single* iterated function. I suspect that if we iterated (c + 1/f(n))^n (with an arbitrary constant c replacing 1 in the original), we would get a whole family of Julia sets plus a Mandelbrot analogue.
If you use a constant whose absolute value is > 1, then an "overlarge" x_n gives you (c + small)^n next, and that will be _large_ rather than _small_, so the nature of the behaviour is different in this case.
(If the absolute value is < 1, maybe that's OK?)
For c=0.5, we get a region kinda like the one for c=1 but with a less crinkly boundary. No big surprises there. For c=0.5 + 0.3i, we get a similar-ish region but its size is hugely increased, with its northern boundary somewhere around 6i. For c=0.8 + 0.6i ... I don't trust the numerics of what I'm doing; it goes rather unstable, not super-surprisingly. For c=0.79 + 0.59i, we _again_ seem to get instabilities. So I think we've found a place where the boundary of the "Mandelbrot set" pokes _inside_ the unit circle. Moving a bit further in in that vicinity, here's c=0.773+0.57i (which doesn't exhibit such instabilities): https://i.imgur.com/sn6Qk4v.png It seems to be disconnected but _not_ dust-like, which you never get with ordinary Julia sets. -- g