Regularity of Embeddings of Infinite-Dimensional Fractal Sets into Finite-Dimensional Spaces

Brian R. Hunt and Vadim Yu. Kaloshin

Nonlinearity 12 (1999), 1263-1275.

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We consider the image of a fractal set $X$ in a Banach space under typical linear and nonlinear projections $\pi$ into $R^N$. 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 H\"older 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 the classical results on preservation of 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 $R^N$, no matter how large $N$ is, the Hausdorff dimension of $\pi(X)$ is less than $d$ (and in fact is less than $2$ no matter how large $d$ is).

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Updated: September 27, 1999