I am wondering whether there exists an algebraic structure(group or modules etc) possess some kind of self-similarity, (sub-group or sub-module have an identical structure with itself) and "irregularity"(undefined) at the same time? if it exists can we find a functor between the category of fractals and this category of algebraic structure in order to study the algebraic properties of fractals?



algebraic structure of fractals
It is a theorem of Douady and Hubbard that the hyperbolic points in the Mandelbrot set form a free noncommutative monoid in a natural way, and that this monoid has a natural action on the whole Mandelbrot set. This is part of a large body of results about the Mandelbrot set that deserve to be better known. I learned about this from Milnor's paper ["Periodic orbits, external rays and the Mandelbrot set"](https://arxiv.org/pdf/math/9905169.pdf) .


