Quantitative measurement of infinite dimensionality - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T01:17:52Z http://mathoverflow.net/feeds/question/59281 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59281/quantitative-measurement-of-infinite-dimensionality Quantitative measurement of infinite dimensionality Benoît Kloeckner 2011-03-23T10:31:27Z 2011-10-15T17:41:33Z <p>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).</p> <p>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.</p> <p>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).</p> <p>The second one, suggested by Greg Kuperberg, is to mimic Hausdorff dimension but replacing the family of "size functions" <code>$(x\mapsto x^s)_s$</code> by another family with similar properties, like $(x\mapsto\exp(-\lambda/x)_\lambda)$.</p> <p>My question is the following: do you know any example of such an invariant in the literature? Where is it used, in what purpose?</p> http://mathoverflow.net/questions/59281/quantitative-measurement-of-infinite-dimensionality/59293#59293 Answer by Bill Johnson for Quantitative measurement of infinite dimensionality Bill Johnson 2011-03-23T12:06:43Z 2011-03-23T12:06:43Z <p>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). </p> <p>For our speculation, see the last paragraph of section 5 in our paper</p> <p>Diamond graphs and super-reflexivity, J. Topology and Analysis 1 (2009), no. 2, 177–189. </p> <p>We have not followed up on this notion and have no idea whether it is good for anything.</p> http://mathoverflow.net/questions/59281/quantitative-measurement-of-infinite-dimensionality/63689#63689 Answer by Benoît Kloeckner for Quantitative measurement of infinite dimensionality Benoît Kloeckner 2011-05-02T09:20:44Z 2011-06-08T17:23:23Z <p>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 href="http://www-fourier.ujf-grenoble.fr/~bkloeckn/papiers/largeness.pdf" rel="nofollow">A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces</a>. Wasserstein spaces were my initial target, while (generalized) Hilbert cubes are handy reference spaces.</p> <p>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 $(x\mapsto \exp(-x^{-s}))_s$.</p> <p>Is it good policy to accept my own answer so that the question is not left open?</p> http://mathoverflow.net/questions/59281/quantitative-measurement-of-infinite-dimensionality/78214#78214 Answer by Gerald Edgar for Quantitative measurement of infinite dimensionality Gerald Edgar 2011-10-15T17:41:33Z 2011-10-15T17:41:33Z <p>See my paper <a href="http://nyjm.albany.edu/j/2007/13-4nf.htm" rel="nofollow">LINK</a></p> <p>Centered densities and fractal measures, New York Journal of Mathematics 13 (2007) 33-87</p> <p>Some references are also at the end of it. In particular, Boardman, Goodey, and McClure.</p>