What real numbers are in the Mandelbrot set?

The Mandelbrot set is defined over the complex numbers and is quite complicated. It's defined by the complex numbers $c$ that remain bounded under the recursion: $$ z_{n+1} = z_n^2 + c,$$ where $z_1 = 0$.

If $c$ is real, then above recursion will remain real. So for what values of $c$ does the recursion remain bounded?

Solution 1:

The Wikipedia page gives the intersection of the set with the real axis as $[-2,0.25]$

Added: You can verify that $-2$ is in the set easily, and that any more negative number decreases each iteration without bound. For the positive end, each iteration is greater than the one before. To hit a limit, you must have $z=z^2+c$, which has the solution $z=\frac{1+\sqrt{1-4c}}2$, which becomes imaginary at $c \gt \frac 14$

Solution 2:

If z is a complex number whose distance to origin is bigger than $|c|$ and 2 then z is a point than scape for the iteration of the function $z^2+c$. It's easy to demostrate this, then the recursion remain bounded inside the closed ball of radio 2, but we can find the mandelbrot set inside $[-2,0.7]\times[-1.2,1.2]$.