| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 753.1 | Sure | CHOVAX::YOUNG | Back from the Shadows Again, | Thu Aug 20 1987 17:01 | 1 |
| Any N where ( -.25 > N > 0 )
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| 753.2 | | CLT::GILBERT | Builder | Thu Aug 20 1987 18:25 | 8 |
| m n
If F (0) = F (0), for any m <> n, then C must be algebraic.
C C
If you can find a trancendental C such that the F series remains
bounded, then the F series cannot be periodic. And such C are easy
to find (based on some common knowledge of the set -- any line segment
wholly in the set provides a multitude of such C).
|
| 753.3 | | PSW::WINALSKI | Paul S. Winalski | Thu Aug 20 1987 18:38 | 6 |
| RE: .0
Oops. I meant to say that C = -2 is an example where the series is bounded and
periodic. C = 2 is not even bounded.
--PSW
|
| 753.4 | | BEING::POSTPISCHIL | Always mount a scratch monkey. | Fri Aug 21 1987 11:43 | 14 |
| Re .1:
> Any N where ( -.25 > N > 0 )
There aren't many positive numbers less than -.25. :-)
Why stop there, though? What is wrong with rational c's such that -1 <
c < 0? They are not periodic because the number of digits after the
radix point (c can be represented with a finite number of digits in
some radix, since it is rational) keeps increasing, and they are
bounded because -1 < z^2+c < 1 as long as -1 < z < 1.
-- edp
|
| 753.5 | ooops | CHOVAX::YOUNG | Back from the Shadows Again, | Fri Aug 21 1987 13:39 | 4 |
| Re.4:
You are right on both counts. I bollixed up a sign when I was
desk checking it.
|
| 753.6 | | ENGINE::ROTH | | Mon Aug 24 1987 11:24 | 20 |
| All the periodic points will lie on the boundry of the Mandelbrot
set. Points exterior to the set diverge, and points interior converge
to a fixed point inside the set.
I've forgotten transformation to use, but you can map the cardioid
shaped main boundry onto a pair of circles - then points at rational
angular points around the circles will be periodic. But so will
points at rational points around the sprouts attached to the main
circle, and so on. Points at irrational angles will exhibit ergodic
motion on the boundry. I forget the expression, but you can also
use number theory to get the points where the filaments fork, and
also points whose angles are transcendental (Liouville numbers) have
some interesting properties.
If you read French, I can send you Hubbard and Douady's paper where
these results were explained and proved. I have not looked at this
subject in a long time, but understood it fairly well when I was
hacking with it...
- Jim
|
| 753.7 | | ENGINE::ROTH | | Mon Aug 24 1987 11:27 | 8 |
| � All the periodic points will lie on the boundry of the Mandelbrot
� set. Points exterior to the set diverge, and points interior converge
� to a fixed point inside the set.
Actually, I should correct this - interior points converge to a
countable set of limit points, not just one.
- Jim
|
| 753.8 | | AITG::DERAMO | Daniel V. {AITG,ZFC}:: D'Eramo | Sun Aug 13 1989 09:56 | 14 |
| What exactly are you saying in .6 and .7?
Is there a countable set {a0, a1, a2, ...} such that for
any interior point c, the sequence 0, c, c^2 + c, ...,
converges to one of the an?
Or, for any interior point c, the set {0, c, c^2 + c, ...}
has countably many limit points?
Or yet another interpretation?
Makes me wish that I read French.
Dan
|