Is there any mathematical proof of fractal self-similarity? When studying fractals, one of the properties named by Benoit Mandelbrot is the self-similarity (and it's variations) of the fractal objects.
From Wikipedia:

In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales.[2] Self-similarity is a typical property of artificial fractals.

This can be easily seen in pictures:
https://upload.wikimedia.org/wikipedia/commons/4/4b/Fractal_fern_explained.png
https://en.wikipedia.org/wiki/Fractal
My question is, is there any way to demonstrate this property in a 'pure mathematical way' ? I know Mandelbrot's purpose was exactly the opposite, to link and demonstrate all this theories in a visual way, but I think any kind of pure demonstration would be great to understand this objects. I haven't been able to find any research about this so I don't know where I could start.

There is no way to demonstrate that property mathematically because the property does not hold. Not all fractals are self-similar and not all self-similar objects are fractals. The rigorous definition of a fractal is an object having a Hausdorff dimension that is less than its topological dimension.
This post asks a similar question and the selected answer gives more detail: Why must fractals be self-referential?
