There are so many fractal which are not uniquely characterize by some fractal parameters like Fractal dimension, Succolarity, Lacunarity, Morphological entropy. Can you suggest some fractal parameters which would be able to characterize distinctly from other fractals?
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5$\begingroup$ Is there even a well-defined definition of fractal to use to universally categorize them in this manner? $\endgroup$– jeremyJun 3, 2010 at 10:07
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6$\begingroup$ @Sk: I strongly disagree with your comment. Jeremy's question is perfectly legitimate and very to-the-point. I have seen several non-equivalent potential definitions of the word "fractal", and I do not believe any of them would find universal acceptance as "the" definition. Your attack on Jeremy's comment strikes me as particularly inappropriate. Even if it had been a naive and unhelpful question (which it most emphatically is not), there is no call for an uncivil response. $\endgroup$– Vaughn ClimenhagaJun 3, 2010 at 15:05
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2$\begingroup$ I'm also not quite sure what to make of the parameters you've listed. I've never heard of succolarity or lacunarity (at least not in this precise context), so a brief explanation in the question would help clarify things. By "morphological entropy" I guess you may mean what I would call "topological entropy", but that assumes the presence of an underlying dynamical system, which you didn't mention. Finally, "fractal dimension" is not a precisely defined concept, but rather a general one that subsumes Hausdorff dimension, box dimension, packing dimension, etc. $\endgroup$– Vaughn ClimenhagaJun 3, 2010 at 15:11
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3$\begingroup$ Abuse of commenters (c.f. "But do not ask such bogus question") is inappropriate. Typing in allcaps is also a sure way to discredit yourself. $\endgroup$– Scott MorrisonJun 3, 2010 at 18:35
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2$\begingroup$ Scott: it looks like the poster hit Caps Lock not Shift at "What I want...". Not that I am sympathetic to his arcument or the question! $\endgroup$– Tom SmithJun 3, 2010 at 22:59
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2 Answers
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Fractal curvature might be the answer. In differential or convex geometry, you need curvature to classify sets up to isometry. So it seems natural to introduce curvature for "fractals" in an attempt to get a finer geometric description. This has been done for mostly self-similar fractals by Winter, Zähle, Rataj, Kombrink, and me (Bohl, formerly Rothe). The full generalization to self-conformal sets is my upcoming PhD thesis.
Philosophically, and literally in differential geometry, curvature takes the second derivative of "coordinates" into account. In contrast, the Hausdorff and packing measures, most other dimensions, Minkowski content (=lacunarity), and surface content are only sensitive to the first derivative. Topological entropy is related to Gibbs / equilibrum measures if you have some kind of iterated function system, and these measures also belong to first order geometry.
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It does matter how you are thinking of fractals. If you are thinking of all sets of Hausdorff dimension that is not integral, then there is essentially no chance of having a number of real-valued invariants that distinguish them. There are always going to be too many sets, I believe. If you use self-similarity (of some kind), or particular ways of generating sets, there is possibly a more reasonable question, of how to find quantitative measures to distinguish them.
answered Jun 3, 2010 at 10:50