> 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](http://vzn1.wordpress.com/code/collatz-conjecture-experiments/) & [Collatz fractal](http://en.wikipedia.org/wiki/Collatz_conjecture#Collatz_fractal)
* (am acknowledging beforehand there is not a strict/widely agreed definition of fractal.)

somewhat related to https://mathoverflow.net/questions/92696/excellent-uses-of-induction-and-recursion