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> what are notable/ prominent inductive proofs relating to fractals?
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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 [Excellent uses of induction and recursion](https://mathoverflow.net/questions/92696/excellent-uses-of-induction-and-recursion)



notable inductive proofs relating to fractals
Spectral decimation is an inductive process where the eigenvalues of a natural Laplacian on "nice" fractals is computed inductively. The idea is that by using a sequence of finite graphs to approximate the graph and graph Laplacians on these graphs one can inductively determine the eigenvalues of the Laplacian on the n+1'st graph from those of the n'th graph. Ultimately the spectrum of the limit Laplacian is the closure of the scaling limit of the spectra on the finite graphs. The original reference for this is Fukushima and Shima "On a spectral analysis for the Sierpiński gasket" from 1992. [MathSciNet](http://www.ams.org/mathscinet-getitem?mr=1245223) or the direct link to the journal is [Springer Pay-wall](http://link.springer.com/article/10.1007/BF00249784). (Apologies to those who do not have access to MathSciNet). 


I am not sure which side of the induction versus iteration divide this lands for you. (Edit: By this divide I am thinking about something like iteratively constructing a fractal level by level but then proving that something is true at all levels. At some point in this process we went from doing something iteratively to doing something inductively. I am not certain that Spectral decimation is a pure enough form of induction for what the OP is asking.) Certainly much of the emphasis on self-similar fractals is an attempt to be able to call upon iterative methods at the very least and in many cases these proofs do become inductive. There is another proof that I am thinking of that occurs in proving Harnack inequalities for harmonic functions on the Sierpinski carpet, but that is for another answer.


The local definition of "nice" for this answer is either post-critically finite fractals or finitely ramified with a doubly transitive symmetry group fractals. 


