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 gn(ε). Recalling that rn = 4n/6, we see

gn+1(4ε) = fn+1(-2 + 4ε/rn+1) = fn+1(-2 + ε/rn) = fn(-2 + ε/rn)2 + (-2 + ε/rn) = gn(ε)2 - 2 + ε/rn

Taking the limit as n → ∞, and letting g(ε) = limn → ∞gn(ε), we obtain

g(4ε) = g(ε)2 - 2

Return to Hurwitz-Robucci scaling.