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Particularly, I'm interested in learning about the dimensions (whether it's always possible to find them, and if so, a concrete way of calculating them) of different types of fractals (given by the Hausdorff dimension, according to a few sources), but particularly iterated function systems, other properties of fractals in general, theorems regarding fractals and their properties, and possibly also open problems regarding fractals. It'd be preferable if linear algebra methods could be used to study fractals. I have a background in undergraduate level analysis, abstract algebra and topology.

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+1 for adding "not just the pretty pictures" –  Hagen von Eitzen Jul 19 '13 at 15:04
    
It might be good to search instead of using the "fractal" keyword, to use instead the "dynamical systems" keywords. Perhaps in particular "complex dynamics". The thing is that with the fractal keyword you will have to be sieving out the less serious publications. –  ABC Jul 19 '13 at 15:07

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The author who would be a great start for looking at fractals constructed by iterated function systems and then studying their features like Hausdorff dimension is Kenneth Falconer. His books from the 90's are a great base to start from. If you want to see what is an amazing use of linear algebra to study functions and Laplacians on fractals Robert Strichartz has a book aimed at exactly your background. The spectral decimation method that Strichartz writes is beautiful when recast into a matrix formulation.

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