I was reading an excellent answer that was given on Quora (found here) that trys to explain why the Mandelbrot contains smaller approximate copies of itself. Here's the section that I'm interested in:
Let $f_c(z) = z^2 + c$, and consider the Taylor series of $f_c^n$ ($f_c$ iterated $n$ times) for some positive integer $n$:
$$f_c^n(z) = f_c^n(0) + \tfrac12(f_c^n)''(0)z^2 + O(z^4)$$ (There are no $z^1$ or $z^3$ terms because $f_c$ is an even function.) If we scale everything by a factor of $k = \tfrac12(f_c^n)''(0)$, we end up with $$k f_c^n\bigl(\tfrac z k\bigr) = \tfrac12(f_c^n)''(0)f_c^n(0) + z^2 + O(z^4)$$ $$= f_{g_n(c)}(z) + O(z^4)$$ where $g_n(c) = \tfrac12(f_c^n)''(0)f_c^n(0)$.
That is, for small $z$, taking $mn$ iterations of $f_c$ works very similarly to taking $m$ iterations of $f_{g_n(c)}$. So we can expect that $g_n(c)$ is in the Mandelbrot set iff $c$ is in something that looks very similar to the Mandelbrot set.
So my question is why is the last paragraph true? Why is there a relation between taking $mn$ iterations of $f_c(z)$ and $m$ iterations of $f_{g_n(c)}(z)$, and why does this show that the Mandelbrot set contains smaller approximate copies of itself?