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


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    $\begingroup$ That's all very fine, but what is a fractal ? Also what is to be considered "iterative sequence of one underlying shape" ? For instance Julia sets (we can agree those enter 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
  • $\begingroup$ @LoïcTeyssier Thanks for quick asking. What I really want to know is if every object of fractal dimension can be constructed by an iterative map regardless of the map being linear or not. $\endgroup$ – Ten Jun 13 '14 at 7:25
  • $\begingroup$ Then I recommend you should rewrite your question based on that definition of a "fractal". Also your "PS" seems irrelevant to me. $\endgroup$ – Loïc Teyssier Jun 13 '14 at 7:28
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    $\begingroup$ As they say, it depends on what you mean by fractal, and also on what you mean by self-similar. $\endgroup$ – Qiaochu Yuan Jun 13 '14 at 7:50
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    $\begingroup$ This depends on what constraints you have on your definition of "map". There is a basic set-theoretic obstruction, namely that there are too many objects fractional dimension to come from things like continuous maps of euclidean space. $\endgroup$ – S. Carnahan Jun 13 '14 at 11:18

No, this is not true. For example, all fractals generated from an Iterated function System, (IFS), are self-similar in the sense that the fractal is the union of the images of the fractal itself, under several maps. See Hutchinson operator.

The classical Julia fractals can be interpreted in this manner, where the two branches of the inverse of $z \to z^2+c$ are used.

Then, there are the random midpoint displacement fractals which are self-similar in a much weaker sense than above, since these fractals involve random choices.

Not all fractals are given by iterating a map (I prefer to call these types of fractals discrete); some fractals are constructed using continuous methods, such as the Lorenz attractor. But note, this fractal can be approximated using iteration of a set of maps.

New Edit: The Mandelbrot fractal cannot (most likely) be constructed via IFS, (I have asked this question before here on MO).

  • $\begingroup$ Thanks for the answer. Please see my reply for Qiaochu Yuan for clarification of the question. $\endgroup$ – Ten Jun 13 '14 at 8:02
  • $\begingroup$ @Ten: Ok, I added a response. $\endgroup$ – Per Alexandersson Jun 13 '14 at 8:08
  • $\begingroup$ Thanks again. So still is it possible to construct any fractal by iteration?(since existence of a construct by continuous method does not preclude the existence of an iterative procedure, unless one gives a proof otherwise) $\endgroup$ – Ten Jun 13 '14 at 9:54
  • $\begingroup$ You need to define what you mean by iteration, and what you mean by map. The midpoint displacement fractal is based on a random process, so there is no iteration of deterministic maps. $\endgroup$ – Per Alexandersson Jun 13 '14 at 10:17
  • $\begingroup$ I have made the question more precise. $\endgroup$ – Ten Jun 13 '14 at 15:08

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