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I'm taking a course on chaotic dynamical systems, and we're talking about attractors with non-integer correlation dimensions, but I can't seem to find a satisfactory definition for this concept. Multiple sources, including Wikipedia, define correlation dimension as the asymptotic order as $\epsilon \rightarrow 0$ of the correlation integral $$ C(\epsilon) = \lim_{N\rightarrow \infty} C_N(\epsilon). $$ The expression under the limit gives a normalized count of the number of pairs of points with mutual distance $<\epsilon$ in a sample of size $N$ from the attractor: $$ C_N(\epsilon) \propto \frac{2}{N(N-1)} \sum_{i=1}^N\sum_{j=1}^N I(| |x_i - x_j|| < \epsilon). $$

But obviously, that count depends on which particular $N$ points are chosen, and is not a function of $N$! This means $C_N(\epsilon)$ is not determined only $N$ and $\epsilon$, and the limit above doesn't make sense!

Of course, even without a rigorous definition, empiricists who study chaotic attractors know how to approximate the correlation dimension of an attractor. They get samples by simulating the trajectory of an initial condition around the attractor. And instead of using the true limits, they just estimate based on the behavior they see for $N$ less than the length of the trajectory they simulated, and $\epsilon$ larger than the granularity at which it sampled the attractor. So if the definition is to conform with what researchers actually estimate, then I think that the quantity under the limit sign should really be the expectation of $C_N(\epsilon)$ when the $x_i$ are sampled from the invariant probability distribution on the attractor. But the only nod to the issue I've read is a single-sentence parenthetical remark on page 848 of Nonlinear Dynamics, by Strogatz:

(Mathematically speaking, the correlation dimension involves an invariant measure supported on a fractal, not just the fractal itself.)

Other than that, as far as I can tell not even the original Grassberger and Procaccia 1983 paper that introduced the concept worried too much about the fact that correlation dimension is not a property of a fractal, but a fractal equipped with a probability distribution from which to sample!

Finally, my question: Is this correct? And why do so many sources that mention correlation use an incomplete definition?

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    $\begingroup$ I don't know much about this, but I do know it's closely related to "Kolmogorov entropy" (which curiously is not mentioned in the Wikipedia article you cited), so maybe including "Kologomorov" as an additional search word/phrase will lead you to something useful. See also the links/cites I give about Kolmogorov entropy in this MSE answer. $\endgroup$ Feb 4, 2023 at 10:45

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