what are notable/ prominent inductive proofs relating to fractals?

the motivation for this question is:

• fractals are very difficult mathematical objects to work with, and many problems/questions about them lie at the boundary of decidable/undecidable (and many are undecidable).
• so it would be useful to study proofs that somehow cleverly/ surprisingly/ successfully "tame" this complexity.
• also, presumably number theory could play a role here, eg a number-theoretic property that has a proof and has also been considered by some to be "fractal" under misc interpretations.
• many famous open problems eg in number theory seem to have a fractal nature and that possibly substantially accounts for their difficulty eg Collatz visualizations/ tiles & Collatz fractal
• (am acknowledging beforehand there is not a strict/widely agreed definition of fractal.)

somewhat related to Excellent uses of induction and recursion