# Can the immediate basin of attraction of super-attracting fixed point at 0 of a polynomial contain non-zero roots?

Let $$f$$ be a polynomial with a super attracting fixed point at $$x=0$$. Can the immediate basin of attraction of the fixed point contain other roots? If so, please provide a specific example with the immediate basin of attraction. If not, why?

• How about $z^3-z^2$? – Mark McClure Feb 24 at 20:20
• @Mark is zero an attracting fixed point? Is, say, $z=1/2$ attracted to zero? – Gerry Myerson Feb 24 at 21:17
• @Mark is 1 in the immediate basin of attraction of 0? It is not obvious – user152801 Feb 24 at 21:35
• what does super attracting mean ? – Piyush Grover Feb 24 at 21:39
• @Piyush A fixed point is super-attracting if the derivative is 0. It means the points in the basin of attraction approach specifically at an exponential rate. In this case, for $z$ in the basin of attraction,$|f(z)-0|\approx b|z-0|^p$ for some constant p. – user152801 Feb 24 at 21:48

Let $$f(z)=az^2(z-1)$$. Zero is superattracting. Now choose $$a$$ so small that $$|f(z)|<|z|/2$$ for $$|z|<2$$. Then the root $$z_0=1$$ is in the immediate domain of attraction.