Competition Idea

[makc]

I was recently looking at actual Mandelbrot orbits and found that, while many orbits clearly diverge or converge, some vibrate for a long time, and some of those some suddenly diverge or converge after significant number of iterations. Does anyone have any idea if this is caused by limited floating point precision, or is it real behavior?

[David Makin]

I was recently looking at actual Mandelbrot orbits and found that, while many orbits clearly diverge or converge, some vibrate for a long time, and some of those some suddenly diverge or converge after significant number of iterations. Does anyone have any idea if this is caused by limited floating point precision, or is it real behavior?

Generally speaking it's real behavior - this can be checked for quite simply - if the behavior changes as the precision used is changed then it's a limitation of the precision used, otherwise it's "real".

[makc]

on the other hand, assuming orbits that start outside the set diverge, and those which start inside converge, what is expected behavior of orbits that start exactly on border? I assume these are stable orbits that remain the same over time, and I see a lot of these (at leas visually so, who knows what would happen if I wait few years). this line of thinking doesnt seem to have any room for orbits that initially converge and then suddenly diverge after long period of time... hence I was thinking it's some kind of accumulating error.

[David Makin]

on the other hand, assuming orbits that start outside the set diverge, and those which start inside converge, what is expected behavior of orbits that start exactly on border? I assume these are stable orbits that remain the same over time, and I see a lot of these (at leas visually so, who knows what would happen if I wait few years). this line of thinking doesnt seem to have any room for orbits that initially converge and then suddenly diverge after long period of time... hence I was thinking it's some kind of accumulating error.

I've found that points very close to the boundary but actually "outside" to exhibit the behavior you describe - at whatever precision - basically the path to infinity is always an exponential one after the final spiral

[kram1032]

a point really exactly on the edge would be -2, as far as I know. It's on the very outest edge of convergence. But it converges just as usual. To 2
-2 -> 2 -> 2 -> 2

[lkmitch]

Points on the boundary generally have unstable orbits.  Consider c = -2 and the standard Mandelbrot set.  The exact sequence is 0, -2, 2, 2, 2, etc.  Computers won't have any difficulty with this since all the values are integers, but if that weren't the case and z got tweaked to 2.000001, the orbit would diverge.  Or if z got tweaked to 1.999999, then the orbit would (probably) be chaotic.  Or, c = 0.25; the orbit converges to z = 0.5.  If z is tweaked to something larger, then the orbit would diverge.  Tips of dendrites are unstable Misiurewicz points (like -2 or (0,1)) and points that are the tangent points between two disks have unstable orbits that straddle the boundary between the periodicity of the parent disk and that of the child disk.

[cKleinhuis]

excuse me, there is no edge , a point is either inside or outside 

[kram1032]

well, anything "more distant" than the edge of the circle with radius 2 in the origin does diverge for sure.
And -2 sits exactly on that edge.

You could find nearly never escaping points by getting closer and closer to the edge (that can't ever be found exactly, true), by putting one point to |z|=2 and one to |z|=1/4, using the same argument for both and then iterating, until it's sure, wether the points converge or diverge and reweight accordingly.
that way, you'd get to some kind of edge after some time

[Timeroot]

Points on the boundary generally have unstable orbits.  Consider c = -2 and the standard Mandelbrot set.  The exact sequence is 0, -2, 2, 2, 2, etc.  Computers won't have any difficulty with this since all the values are integers, but if that weren't the case and z got tweaked to 2.000001, the orbit would diverge.  Or if z got tweaked to 1.999999, then the orbit would (probably) be chaotic.  Or, c = 0.25; the orbit converges to z = 0.5.  If z is tweaked to something larger, then the orbit would diverge.  Tips of dendrites are unstable Misiurewicz points (like -2 or (0,1)) and points that are the tangent points between two disks have unstable orbits that straddle the boundary between the periodicity of the parent disk and that of the child disk.

...and the one other case so often forgotten: strange attractors. A point the doesn't map into an unstable orbit (Misiurewicz points) and aren't parabolic (connecting two disks) exhibit even weirder behavior; if memory server, (-0.8,0.2) is one such point.

EDIT: Sorry, that should be (-0.8,0.15). A similar is (-1.2,0.15). Another, "different" kind of strange point is the Grossman-Whatshisname band-merging point. I think one of the banners is actually a picture of it. If you zoom towards the p-3 minibrot, then back towards the p-5 minibrot, then back to the p-7, etc. you'll find they converge (alternating) to one point. It's also not a Misiurwecz point, I think, nor is parabolic. The MSet zoo is infinite, it seems..

[lkmitch]

excuse me, there is no edge , a point is either inside or outside 

It's true that a point is either inside or outside, the inside is composed of two sets, the interior and the boundary.  An interior point is one such that a circle of positive radius can be constructed centered on the interior point such that all of the points inside the circle are inside the set.  A boundary point is one such that any circle centered on the point, no matter how small the radius, will contain both inside and outside points.

Here is a way to find some edge/boundary points:  Let theta be an angle in radians, whose value is a rational number (like 1.5 radians).  Then, r = (1 - cos(theta))/2, x = r cos(theta) + 0.25, and y = r sin(theta).  The point c = x+iy is on the boundary of the main cardioid of the Mandelbrot set, but since theta is not a fraction times 2pi, c is not a tangent point of a disk.

[kram1032]

So, both .25 and -2 are boundary points, right?
Both of them do converge but shifting them just a tiny bit away from the center will make them diverge, so any circle around them will have both types of points.

However, those two are most likely the least interesting boundary points - What's about the outer most boundary on the imaginary values, for instance? (However, that was discussed in an other thread, I think...)