Attempt: If $x_-^2+c=x_-$ then $x_-=\dfrac{1-\sqrt{1-4c}}{2}$ which is attracting for $|f(x)|<1$ i.e $-2<c<\dfrac14$. How do I find the set of points $x$ such that the orbit $f^n(x) \to x_-$ (where $f^n(x)$ is $f$ applied $n$ times)?
2025-01-13 02:32:24.1736735544
What is the basin of attraction for the attracting fixed point $x_-$ of $f(x) = x^2+c$
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For these $c$, the basin of attraction in the complex plane is a connected filled Julia set and so the basin of attraction in the real line is an interval.
One of the extremes of this interval is the other fixed point, the repelling fixed point $x_+$. The other extreme is sent to this fixed point by $f$; it is easily seen that this point is $-x_+$.