Self Similarity in fractals

Question

I'm looking into strict self similarity in fractals as part of a project for first year an from my understanding of the topic the Julia set is strictly self similar. However I've not been able to find anywhere this is firmly stated and wanted to check before including it in my work as fact. Thank you

Answer 1

Julia sets are not generally strictly self-similar. They often display a loose degree of self-similarity, though, because they can be viewed as the invariant set of something like an IFS.

For example, if you're working with the function $f_c(z)=z^2+c$, then the Julia set $J_c$ is invariant under the action of the pair of functions $\pm\sqrt{z-c}$. That is,

$$J_c = \sqrt{J_c-c} \; \bigcup\; -\sqrt{J_c-c}.$$

Here's an illustration for $f(z)=z^2-1$.