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Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain cases, this lead to new manifolds which could not be realized as subsets of euclidean spaces. Eg non-orientable surfaces cannot be embedded in $R^3$, non-algebraic manifolds etc.

Nowadays, fractals are found as iterations of certain polynomial maps on euclidean space. (or some other ways, as wikipedia says, but still embdedded). Are we missing out on some fractals? Is there an intrinsic definition of fractals? (This would also have to come with an intrinsic definition of dimension. Haudsorff dimension is still an embedded dimensions, using covering of balls in euclidean space.)

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    $\begingroup$ All smooth manifolds can be realized as subsets of Euclidean space of a suitable dimension. $\endgroup$
    – Qfwfq
    Commented May 18, 2010 at 14:14
  • $\begingroup$ I believe you need the addition there of "locally". Any open ball can be mapped to an open set in Euclidean space $\endgroup$
    – Michael Hoffman
    Commented May 18, 2010 at 16:46
  • $\begingroup$ @Qfwfq That is true, due to Nash. But fractals are far from being smooth manifolds. $\endgroup$
    – j0equ1nn
    Commented Jun 1, 2018 at 1:28
  • $\begingroup$ @j0equ1nn: it was in response to the sentence "And in certain cases, this lead to new manifolds which could not be realized as subsets of euclidean spaces". I interpreted it as the OP asserting that some manifolds can't be embedded in euclidean spaces (which is false). But perhaps the OP just meant that certain manifolds can't be embedded in a specific euclidean space (which is the case for the Klein bottle and $\mathbb{R}^3$, mentioned in the first paragraph of the question)... $\endgroup$
    – Qfwfq
    Commented Jun 1, 2018 at 17:18

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Topological dimension (say, covering dimension) $\dim_\mathrm{T}$ and Hausdorff dimension $\dim_\mathrm{H}$ both make sense for metric spaces. Benoit Mandelbrot defined $A$ to be a fractal iff $\dim_\mathrm{T} A < \dim_\mathrm{H} A$. The packing dimension $\dim_\mathrm{P}$ also makes sense in metric space. James Taylor defined $A$ to be a fractal iff $\dim_\mathrm{H} A = \dim_\mathrm{P} A$. Also making sense for metric space is the definition of Michael Barnsley ... A fractal is an element of the hyperspace $\mathbb{H}(K)$ of a compact metric space $K$. Perhaps you have your own, different, definition?

Definitions for all these are in my book Integral, Probability, and Fractal Measures

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  • $\begingroup$ thanks edgar. and all these definitions of dimension are invariant under isometry/hemeomorphism? $\endgroup$
    – user2529
    Commented May 19, 2010 at 3:01
  • $\begingroup$ for instance any simply connected proper domain of the plane (which could be a fractal) is biholomorphic to the open disc (which is not a fractal). but yet biholomorphism implies homeomorphism. $\endgroup$
    – user2529
    Commented May 19, 2010 at 3:06
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    $\begingroup$ @Colin: I prefer to say not that the domain is fractal, but the boundary is. And that biholomorphism generally is not Lipschitz on the boundary. $\endgroup$
    – Gerald Edgar
    Commented May 19, 2010 at 14:40
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To the best of my knowledge there is no universally agreed upon precise definition of the word "fractal", so it's not clear to me exactly what would or would not constitute an example of a fractal that is not embedded in Euclidean space.

However, the various quantities referred to as "fractal dimension" -- Hausdorff dimension, box dimension, etc. -- do not actually require an ambient Euclidean space for their definition. All you need is a metric on the set X under consideration -- this is enough to define "balls of radius r", and once you can do that the definition of Hausdorff dimension, box dimension, etc. goes through exactly as in the Euclidean case.

In fact, there's a very general framework for all these dimensional quantities (for me the standard reference is "Dimension Theory in Dynamical Systems" by Yakov Pesin), which can be formulated in a setting completely independent of Euclidean space.

As a possible example of a "non-Euclidean fractal", I would consider the symbolic space $\Sigma_2^+ = \{0,1\}^\mathbb{N}$ with a metric such as $d(x,y) = 2^{-t(x,y)}$, where $t(x,y)$ is the first coordinate in which x and y differ. This is homeomorphic to the Cantor set but not embedded in Euclidean space.

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Let me comment briefly on the notion of "fractal" (although we really don't need one, do we?). More precisely, let me comment on the one based on Hausdorff and packing dimensions. True that packing dimension occurs naturally on the boundary of some special Kleinian groups, but that's it right? In other words, can you provide other "natural" occurrences of packing dimension and/or packing "measure" (hopefully many of them). If not, then why should we have a definition involving such a notion? A notion should be somewhat motivated, say in the context of dynamics or of some other area. Let me note that Hausdorff measure and Hausdorff dimension seem to occur "naturally" much more.

Incidentally, I usually recommend Pesin's ICM address as a departure point (notice that it appeared earlier than the larger Russian Math. Surveys paper, as always covering almost all paths as is his style). Pesin's book is a quite important reference, but it distracts the reader with its more technical first part. So, for the noninitiated his wonderful book is really not the best reference.

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