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?
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2$\begingroup$ You should check out the following book: en.wikipedia.org/wiki/Indra%27s_Pearls_(book) $\endgroup$– Per AlexanderssonCommented Jun 2, 2018 at 10:28
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2$\begingroup$ Also, arxiv.org/pdf/1112.5415.pdf might be of interest $\endgroup$– Per AlexanderssonCommented Jun 2, 2018 at 10:30
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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" .
answered Jun 2, 2018 at 11:18