I have been looking for the proof that for any z that is part of the mandelbrot set so is its conjugate meaning reflection in the x axis. can anyone show this in detail? Thank you in advance!
2026-04-01 17:26:31.1775064391
Proof of Mandelbrot Set Reflection on the x axis
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The Mandelbrot iteration is $$z_{c,n+1}=z_{c,n}^2+c,z_{c,0}=0$$ We clearly have $z_{c,0}=0=z_{c^*,0}^*$ for all $c$. Now suppose $z_{c,k}=z_{c^*,k}^*$ for some $k$. Then $$z_{c,k+1}=z_{c,k}^2+c=(z_{c^*,k}^*)^2+c$$ $$=(z_{c^*,k}^2)^*+(c^*)^*=(z_{c^*,k}^2+c^*)^*=z_{c^*,k+1}^*$$ So, by induction, $z_{c,n}$ and $z_{c^*,n}$ have trajectories conjugate of each other, which means they both diverge or both stay bounded, and the Mandelbrot set is symmetric on the real axis.