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Brian R. Hunt and Vadim Yu. Kaloshin

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Nonlinearity 12 (1999), 1263-1275.
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Online abstract and download information

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*