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In Chaos Theory, the Correlation Dimension is defined to calculate the dimension of fractals.

At least in the context where I've learnt it, it is applied to fractals made up of sets of points.

Is it also used to define fractals that are not sets of points, such as von Koch curve and Sierpinski carpet?

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Yes, you can calculate it for any object.

In introductory courses, it might be common to see the definition being applied to simpler cases, and an important application of it is indeed the characterization of finite time series - but this measure actually was introduced precisely for investigating general attractors in finite- and infinite-dimensional systems.

When the object is not made up from a finite number of points, some sort of sampling must be performed. From the paper by Grassberger and Procacia (1983):

The measure [...] is obtained from the correlations between random points on the attractor [...] obtained e.g. from a time series

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  • $\begingroup$ So this method can be applied to von Koch curve and Sierpinski carpet as well, by, for instance, generating random points on them? $\endgroup$ – The Notorious Dec 21 '18 at 5:11
  • $\begingroup$ @The Notorious Exactly. $\endgroup$ – stafusa Dec 21 '18 at 6:43

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