The Mandelbrot Set and Julia Sets
Scalings in the Mandelbrot Set
Hurwitz-Robucci scaling -
Universal Function Scaling Relation
Now we find a scaling relation for the g
n
(ε).
Recalling
that r
n
= 4
n
/6, we see
g
n+1
(4ε)
= f
n+1
(-2 + 4ε/r
n+1
)
= f
n+1
(-2 + ε/r
n
)
= f
n
(-2 + ε/r
n
)
2
+ (-2 + ε/r
n
)
= g
n
(ε)
2
- 2 + ε/r
n
Taking the limit as n → ∞, and letting g(ε) = lim
n → ∞
g
n
(ε), we obtain
g(4ε) = g(ε)
2
- 2
Return to
Hurwitz-Robucci scaling
.