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Coming from a specific field in algebraic geometry, I am a total noob in Fractal Theory and I'd like to learn it a bit. I hope I am tolerated for my maybe-trivial questions. I just read about the Weierstrass-Mandelbrot fractal (it's also simply called Weierstrass fractal using the Weierstrass function.. but there are dozens of Weierstrass functions so I'd rather call it "Weierstrass-Mandelbrot" function). The definition of this fractal is found in Wikipedia. I got easily impressed by it.

My question is whether there are nowhere differentiable continuous functions (between real numbers) whose graphs are not fractals? Is the WM function the easiest example of a nowhere differentiable continuous function?

The other question is quite basic (for experts probably). I have seen the definition of fractal in Wikipedia. This definition uses self-similarity. But in a reference of mine (from a lecture note) I get a definition that makes use of an inequality with Hausdorff-dimension and inductive dimension. Are these definitions equivalent or is the precise definition still under debate? (My reference suggests that the suggested definition was the former definition by Mandelbrot, and then this definition was changed as Mandelbrot fractals don't follow this definition.) A little enlightening would help :)

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    $\begingroup$ there, now it is really fixed. =) $\endgroup$
    – Willie Wong
    Commented Jul 13, 2011 at 11:12
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    $\begingroup$ Among the continuous functions on [0,1], the functions that are not differentiable at any point are topologically generic (classical result going back to Banach and Mazurkiewicz), and also prevalent, which is a measure-theoretical genericity valid in infinite dimensional spaces (this is due to Hunt). Clearly, a generic continuous function won't have any self-similarity whatsoever. When speaking of curves, though, nowhere differentiability is considered a hallmark of fractality. As others said, though, fractal is a vague concept which does not and cannot have a precise definition. $\endgroup$
    – Pablo Shmerkin
    Commented Jul 13, 2011 at 17:37

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My question is whether there are nowhere differentiable continuous functions (between real numbers) whose graph are not fractals?

Of course this depends on your definition of fractal. There are nowhere-differentiable functions with graph of Hausdorff dimension 1.

Is the WM function the easiest example of a nowhere differentiable continuous function?

No.
For example, a nowhere-differentiable function due to Kießwetter was designed to be used with high-school students in Germany. English translation in my book: Classics on Fractals

Are these definitions equivalent?

No, the definition with self-similarity is not equivalent to Hausdorff dimension > topological dimension. [Using self-similarity as a definition of fractal should be considered something to use for non-mathematicians who are curious about the subject, but have no hope to understand measures and such for the real definition.]

Is the precise definition still under debate?

Mandelbrot gave the definition: Hausdorff dimension strictly greater than topological dimension. He later wrote that he regretted this, and instead it should be left undefined. Others have provided other definitions. For actual mathematical papers, the authors of course state what they are proving in real mathematical language, not using the word fractal or just using it for the vague explanatory part of the paper.

Added:
Kiesswetter function, two figures from Classics on Fractals

Figure 18.2 (source: Wayback Machine)

     Figure 18.3 (source: Wayback Machine)

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  • $\begingroup$ Thanks for the detailed info. Do you happen to have a pictorial link to the Kießwetter fractal? I wasn't able to find it by doing a google image search. $\endgroup$
    – Jose Capco
    Commented Jul 14, 2011 at 8:18
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    – Gerald Edgar
    Commented Jul 14, 2011 at 14:48
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A quick, partial answer to your second question about the definition of fractals. If a fractal is generated by an iterated function system with a scaling ratio less than one then you do get a Hausdorff dimension less than the inductive dimension. However it is not particularly difficult to create a set with Hausdorff dimension less than inductive dimension that should be a fractal that isn't self-similar. The idea is to choose between two iterated function systems aperiodically.

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You can also create a continuous, non-differentiable function by restricting the height map produced by the midpoint displacement algorithm to a line: http://en.wikipedia.org/wiki/Diamond-square_algorithm

This will not be self-similar, each open segment will with probability 1 be unique (if I am not mistaken). The displacement factor can be tuned so that the fractal dimension is any number between 1 and 2.

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The question "are nowhere differentiable continuous functions (between real numbers) whose graph are not fractals?" has no answer, because there is no universally accepted definition of fractal:

https://mathoverflow.net/questions/56677#56779

But I'd say yes :   Every nowhere differentiable C⁰ function has some "fractalness".

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  1. The word "fractal" has no established commonly accepted definition. (Some definitions involve self-similarity, others only Hausdorff dimension).

  2. You should specify what exactly you mean by Weierstrass-Mandelbrot. There are several of constructions of continuous nowhere differentiable functions. Probably the simplest one is due to van der Waerden. It is given in Landau's famous Calculus textbook immediately after the definition of the derivative.

  3. The exact value of Hausdorff dimension of the original Weierstrass graph is not known (at least to me). Some partial results can be found in

MR1002918 Przytycki, F.; Urbański, M. On the Hausdorff dimension of some fractal sets. Studia Math. 93 (1989), no. 2, 155–186.

  1. Certainly, there are many nowhere differentiable functions whose graph is not a fractal in any accepted sense.
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