Well the question is as simple as that and what I really want to see is if there is a mathematical proof that can tell whether every fractal(object of any non-integer dimension) can be constructed by iterated function systems(such as in p.55 of this). A reference will be very helpful.

EDIT: 1) changed 'iterated maps' -> iterated functions systems 2) a similar question was asked here 3) An article suggesting a counter-example is suggested in the comments below

whatis a fractal ? Also what is to be considered "iterative sequence of one underlying shape" ? For instance Julia sets (we can agreethoseenter the "fractal" realm) are not affine self-similar, but they do have some non-linear self-similarity properties. Where do you draw the line ? I'm afraid you won't be able to phrase your question in a mathematically precise way... $\endgroup$ – Loïc Teyssier Jun 13 '14 at 7:15