# Are the mini Mandelbrots on the X-axis exact copies of the Mandelbrot set?

Although not all of the mini Mandelbrots exact copies of the whole set (for example, look at this question), are the mini Mandelbrots on the X-axis exact copies of the whole set? More specifically, if we shrink the entire Mandelbrot set and move it to the location of any of these mini Mandelbrots, will it perfectly overlap onto them? (Obviously not in a trivial way like shrinking it almost completely and then putting it inside the cardioid of one of them...)

If so, is there any proof for this?

Thanks!

• No, they are not exact copies. Oct 26, 2022 at 23:22
• @GEdgar I tried to confirm on some and you're correct, thanks! Oct 26, 2022 at 23:48
• No mini Mandelbrot within the Mandelbrot set is an exact copy of the whole Mandelbrot set, because there is only one perfect circle in the whole Mandelbrot set. See math.stackexchange.com/questions/1857237/… Oct 27, 2022 at 13:39

As pointed out, they are not exact copies.

For example, here is an overlay of two parts, and clearly they don't perfectly overlap: According to the journal article here, the period $$3$$ hyperbolic components are given by

$$c=-\frac{7}{4}-\frac{20}{9} \left[ \sinh \left( \omega (z)+\frac{2k\pi i}{3} \right)-\frac{1}{4\sqrt{5}} \right]^2$$

$$\omega (z)=\frac{1}{3}{\operatorname {Arcsinh}} \left( \frac{88 - 27z}{80\sqrt{5}} \right)$$

Expanding numerically into Fourier series in $$z=e^{i\theta}$$ for $$k=0$$:

$$c=-1.754877666+0.009517759z(1-0.461468994z-0.026854399z^2+\ldots)$$

which is not a perfect cardioid.

Compare with the hyperbolic components for periods $$1$$ and $$2$$ here.