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I recently encountered the metric mean dimension, which is a numerical metric invariant of (discrete time, compact space) dynamical systems that refines topological entropy for infinite-entropy systems. I am wondering if anything similar can be found in the literature for any metric notion of dimension (Let say that by ``metric'' means bi-Lipschitz invariant).

Put another way, I have a compact metric space $X$ that has infinite dimension for any sensible notion of dimension, and I would like to make this statement quantitative. I see two ways to do this.

The first one is to mimic the box-dimension, and consider the (extra-polynomial) growth rate of the smallest number of $\varepsilon$-balls needed to cover $X$ when $\varepsilon^{-1}$ goes to infinity. This is the simplest way to go, but I am concerned by the fact that box dimension have not as nice a behavior than Hausdorff dimension (for example countable spaces can have positive box dimension).

The second one, suggested by Greg Kuperberg, is to mimic Hausdorff dimension but replacing the family of "size functions" $(x\mapsto x^s)_s$ by another family with similar properties, like $(x\mapsto\exp(-\lambda/x)_\lambda)$.

My question is the following: do you know any example of such an invariant in the literature? Where is it used, in what purpose?

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Gideon Schechtman and I speculated on a notion of dimension (we call it complexity) of a general metric space that comes from the theory of Lipschitz $p$-summing operators that Farmer and I introduced. A metric space has finite complexity provided the Lipschitz $1$-summing norm of the identity function on the space is finite. For an infinite set with all distances one, which we consider a simple metric space, the Lipschitz $1$-summing norm of the identity is two. For $\mathbb{R}^n$, this parameter is about $n^{1/2}$. When the Lipschitz $1$-summing norm of the identity is infinite, the asymptotics of the Lipschitz $(p,1)$-summing norm of the identity as $p$ decreases to one describes the complexity of the space (the point being that for $p>1$, this parameter is always finite and tends to the Lipschitz $1$-summing norm when $p$ decreases to one).

For our speculation, see the last paragraph of section 5 in our paper

Diamond graphs and super-reflexivity, J. Topology and Analysis 1 (2009), no. 2, 177–189.

We have not followed up on this notion and have no idea whether it is good for anything.

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Thanks ; this construction seems however more difficult to interpret than the more or less straightforward examples I gave. I don't feel how difficult it is to estimate these summing norms. – Benoît Kloeckner Mar 25 '11 at 12:35
It is certainly more difficult to understand than the examples you mentioned, and it is difficult to compute for many examples. It is however quite natural when you come into metric geometry from geometric functional analysis. – Bill Johnson Mar 26 '11 at 12:05
up vote 2 down vote accepted

After failing to find any evidence that the notions I asked for have been previously defined, I chose to write things down. The resulting paper is available: A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces. Wasserstein spaces were my initial target, while (generalized) Hilbert cubes are handy reference spaces.

By the way, I should stress that using the family of functions $(x\mapsto \exp(-\lambda/x))_\lambda$ suggested in the question is a bad idea: the resulting analogue to Hausdorff dimension is not bi-Lipschitz invariant. One has to use cruder families like $(x\mapsto \exp(-x^{-s}))_s$.

Is it good policy to accept my own answer so that the question is not left open?

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I don't see a problem with accepting your own answer. – S. Carnahan Sep 26 '11 at 16:35

See my paper LINK

Centered densities and fractal measures, New York Journal of Mathematics 13 (2007) 33-87

Some references are also at the end of it. In particular, Boardman, Goodey, and McClure.

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Thanks for the references. I find somewhat surprising that so much has been done for generalized Hausdorff measures and so little for generalized Hausdorff dimensions. – Benoît Kloeckner Oct 16 '11 at 9:36

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