Hunt, Brian R.; Kaloshin, Vadim Yu Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. (English) Zbl 0932.28006 Nonlinearity 12, No. 5, 1263-1275 (1999). 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). Cited in 6 ReviewsCited in 62 Documents 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 Keywords:image of a fractal set; Banach space; linear and nonlinear projections; box-counting dimension; Hölder exponent; Hausdorff dimension PDFBibTeX XMLCite \textit{B. R. Hunt} and \textit{V. Y. Kaloshin}, Nonlinearity 12, No. 5, 1263--1275 (1999; Zbl 0932.28006) Full Text: DOI Link