Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces

doi:10.1088/0951-7715/12/5/303

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Nonlinearity

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Nonlinearity Volume 12 Number 5 Create an alert RSS this journal

Brian R Hunt and Vadim Yu Kaloshin 1999 Nonlinearity 12 1263 doi:10.1088/0951-7715/12/5/303

Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces

Brian R Hunt and Vadim Yu Kaloshin

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Recommended by Professor P Cvitanovic

We consider the image of a fractal set X in a Banach space under typical linear and nonlinear projections into ^N. We prove that when N exceeds twice the box-counting dimension of X, then almost every (in the sense of prevalence) such is one-to-one on X, and we give an explicit bound on the Hölder exponent of the inverse of the restriction of 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 into ^N, no matter how large N is, the Hausdorff dimension of (X) is less than d (and in fact, is less than two, no matter how large d is).