Types of points in the parametric plane
Types of points in the parametric plane :
- exterior of M-set (Outside set) = basin of attraction of infinity = Orbit diverges to infinity = unbounded orbits
- M-set (bounded orbits), periodic and preperiodic points
- interior of M-set (Inside set, not on boundary: Orbit converges to a finite attracting point or cycle)
- border of M-set
- Myrberg-Feingenbaum point
with 2 ( transcendental) irrational external arguments
- Misiurewicz points (tip or branch of filament
Orbit converges on finite repelling cycle);
preperiodic point,
Phi is non-periodic but preperiodic under Fc
point has one ( tips) or more (branch) rational external arguments with even denominator, preperiodic under doubling map.
examples:
- tip of the main antenna c=-2 ; external angle=1/2;
- roots of components
It may be bud attachment point (point by wich a component is attached to bigger componenet) or cusp
internal angle(root) = 0
periodic orbit is rationally indifffrent = parabolic, periodic point
The disc attached to the main cardioid at internal angle m/n have periodic orbit of cycle = n.
The root point have 2 external arguments ( rational with odd denominator, periodic with period n under doubling map)= 2 external rays land on root point
If it is a primitive ( ( not budding from another one) component then root is the cusp of the cardioid.
Julia sets for parabolic parameters
cusp of the cardioid:
- cusp of main cardioid c=1/4
- cusps in wich filament arrives
- other filament points, with irrational external angle: Chaotic orbit
- other points of component's border, not on a cusp, perhaps at the limit of an alternating series of buds, with irrational external angle: Orbit chaotic. (Julia set has Siegel disks.)
Sources:
Main page
Autor: Adam Majewski
adammaj1-at-o2-dot--pl
Feel free to e-mail me. (:-))
About