How would I find the filled Julia set for $f(z)=z^3$? I know it should be the filled unit circle, but I don't quite understand the math. This is what I have so far: Fixed points $z^3=z$ so $z=1,-1,0,\infty$ $f'(1)=3>1$--repelling $f'(-1)=3>1$--repelling $f'(0)=0<1$--super attracting $f'(\infty)=0<1$--super attracting
One definition says that the filled Julia set is the set of points that have bounded orbit.
It is clear that $z$ has bounded orbit under $f$ iff $|z|\le1$.