Topological Conjugate
If h(x) with inverse hInv(x) is a bijective function (one-to-one), then f(x) = h( g( hInv(x) ) )is the topological conjugate of g(x).
My goal is: if g(x) is the generation function of a perfect fractal set, then f(x) should be a perfect fractal set.
Perfect fractal set: a fractal set with infinite small apple manikins.
In my tests I have found that the bijectivity of h is not always necessary for that goal. E.g. h(x) = x^2.
Variants
If the generation function is of the form g(x) + c, then f(x) + c = h( g (hInv(x) ) ) + c.In general the generation function is of the form g(x,c), then f(x,c) = h( g (hInv(x),c ) ).
Sometimes f(x,c) = h( g (hInv(x),c ) ) + c is working, too, I don't know why.
Outer Scale and Inner Scale
A generation function f(x,c) can be scaled by a variable: scale*f(x,c).A topological conjugate has two scales: outerscale*f(x,c) = outerscale*h( innerscale*g (hInv(x),c ) ).
Outer scale and inner scale is mentioned here, because these terms are used in the pictures section.