Fractal animation.

You'd have to be more specific what you actually mean by "fractal" , but AE's own Fractal effect (classic Mandelbrot) would be a start. There's a similar effect in the Sapphire suite as well. Otherwise of course consider tools like Flame Painter, Mandelbulb and so on. Pretty much any of these "fractal painting" programs have options to output image sequences or clips which then could be processed in AE. Mirroring can be done with CC Kaleida, Motion Tile, Mirror and an assortment of manual techniques slicing & dicing pre-compositions and arranging them in patterns. Really depends. An example image of the style you're looking for would certainly help.

Mylenium 

I'm not sure if this is specific enough for what you're looking for, but this is pretty straightforward.
1. Create a new solid layer (CMD/CTRL + Y)
2. Add the effect Generate > Fractal

3. Change settings to suit your needs.

If you need more control over the way this is reflected, the Mirror and/or CC Kaleida effects may also be helpful.