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 🙂
Best Answer
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