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Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. (English) Zbl 0932.28006

Summary: We consider the image of a fractal set \(X\) in a Banach space under typical linear and nonlinear projections \(\pi\) into \(\mathbb{R}^N\). We prove that when \(N\) exceeds twice the box-counting dimension of \(X\), then almost every (in the sense of prevalence) such \(\pi\) is one-to-one on \(X\), and we give an explicit bound on the Hölder exponent of the inverse of the restriction of \(\pi\) to \(X\). The same quantity also bounds the factor by which the Hausdorff dimension of \(X\) can decrease under these projections. Such a bound is motivated by our discovery that the Hausdorff dimension of \(X\) need not be preserved by typical projections, in contrast to classical results on the preservation of a Hausdorff dimension by projections between finite-dimensional spaces. We give an example for any positive number \(d\) of a set \(X\) with box-counting and Hausdorff dimension \(d\) in the real Hilbert space \(\ell^2\) such that for all projections \(\pi\) into \(\mathbb{R}^N\), no matter how large \(N\) is, the Hausdorff dimension of \(\pi(X)\) is less than \(d\) (and in fact, is less than two, no matter how large \(d\) is).

MSC:

28A80 Fractals
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
94A17 Measures of information, entropy
60B05 Probability measures on topological spaces
28A78 Hausdorff and packing measures
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