The Mandelbrot Set and Julia Sets
Scalings in the Mandelbrot Set - Component Diameter
How can we measure the sizes of the 2-cycle, 4-cycle,
8-cycle, 16-cycle, ... components of the
period-doubling cascade?
The
diameter of the 2-cycle component is the distance between where that component
attaches to the big cardioid, and where the 4-cycle component attaches to that
component.
Similarly, we can think of the diameter of the 4-cycle
component as the distance between where the 4-cycle component attaches to the
2-cycle component, and where the 8-cycle component attaches to
the 4-cycle component.
The same kind of measurement can be made all along the
period-doubling cascade.
Call c1 the point of attachment of the 2-cycle and
1-cycle components.
Call c2 the point of attachment of the 4-cycle
and 2-cycle components.
Call c3 the point of attachemnt of the 8-cycle
and 4-cycle components.
In general, call cn the point of attachment
of the 2n-cycle and 2n-1-cycle components
in the period-doubling cascade.
Careful numerical experiments give these values:
| c1 = -0.75 |
| c2 = -1.25 |
| c3 = -1.3680989394 ... |
| c4 = -1.3940461566 ... |
| c5 = -1.3996312389 ... |
| c6 = -1.4008287424 ... |
| c7 = -1.4010852713 ... |
| c8 = -1.401140214699 ... |
| c9 = -1.401151982029 ... |
| c10 = -1.401154502237 ... |
| ... |
The diameter of the 2n-cycle component in this cascade is
|cn+1 - cn|.
Return to Scalings in the Mandelbrot set.
