Re: Is Mandelbrot Set symmetrical around the imaginary axis?

Thanks for the correction in the demonstration, it's correct now.

>I can not see that the Julia polynomial has only even powers.
>Could you spell it out please?

Well, basically it happens because we are squaring all the time. Look:

z0 = p0(z) = z
z1 = p1(z) = z0^2 + c = (z^ 2) + c
z2 = p2(z) = z1^2 + c = (z^ 4) + (z^2)2c + c^2
z3 = p3(z) = z2^2 + c = (z^ 8) + (z^6)4c + (z^4)6c^2 + (z^2)4c^3 +
(z^0)c^4
z4 = p4(z) = z3^2 + c = (z^16) + ......

As you see only even powers can happen. Thus, the complete polinomial
evaluates to the same value regardles the sign of z (in complex numbers
z^2=(-z)^2 too as in real numbers). As opposite to the Mandelbrot set,
the symmetry happens not only in the modulus |zn|, but in the argument
too. This means that the complete dynamics are really symetrical under
180 degree rotation.



>How can you prove that Julias at either sides of Mandel have opposite spins?


Hmm, I think I found something. I'm not a mathematician, so I hope not
to do too severe mistakes... We must use the two following properties:

1] Let "z" be a complex number and "u" a natural number. Then (z')^u =
(z^u)'
2] z1 * z2' = ( z1' * z2)'

>>From the polinomial expansion of orbit aboce, pn(z), we can see that
the final point (after n iterations) is just the result of a power
series.

pn(z,c) = sum { Ak*Z^(2k) }

where the coefficients Ak are just of the form a*c^b.

Now, when we change the parameter "c" from one side of the Mandelbrot
set to the other, we are basically conjugating c. Let's see what
happens to pn(z,c'):

pn(z,c') = sum { Ak' * Z^(2k) }

and from the properties [1] and [2],

pn(z,c') = sum { (Ak * Z^(2k)')' } = (sum { Ak * Z^(2k)' } )'

Now, the Julia set J(c') depends on |pn(z,c')|, so

|pn(z,c')| = | (sum { Ak * Z^(2k)' } )' | = | sum { Ak * Z^(2k)' } |

Again, by property [1] we see that

|pn(z,c')| = |pn(z',c)|

So, basically, flipping/conjugating "c" in the parameter plane (where
Mandelbrot set lives) means mirroring the Julia set. In other words, a
Julia set living in a certain "side" of the real axis looks the same as
his "twin" in the other side, but with a symmetry along this axis.

Sorry to the mathematicians for the way I "soiled" the maths with these
"half-demonstrations".



Iñigo Quilez

.

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