Here's an excerpt from the book:
The highly intricate structure of the Julia set illustrated in Figure 0.6 stems from the single quadratic function $f(z) = z^2 + c$ for a suitable constant $c$. Although the set is not strictly self-similar in the sense that the Cantor set and von Koch curve are, it is quasi-self-similar in that arbitrarily small portions of the set can be magnified and then distorted smoothly to coincide with a large part of the set.
I don't know if the author rigorously defines quasi-self-similarity later in the text, but the "definition" above seems vague. In the particular example of the Julia set, what is meant by "arbitrarily small portions of the set can be magnified and then distorted smoothly to coincide with a large part of the set"? What exactly are these distortions, and could someone help me understand with pictures how the magnified part on distortion coincides with a part of the set? How much do I need to magnify, and what part?
Most of these questions are easy to answer in the case of Cantor sets, von Koch curve, and the Sierpinski triangle - since these are self-similar fractals. Quasi-self-similarity isn't that easy a nut to crack, it seems.
I also came across this animation for $f(z) = z^2 - 1$, but I'm not sure what it's trying to convey.
References:
Falconer, Fractal Geometry.