How Projections Affect the Dimension Spectrum of Fractal Measures

Brian R. Hunt and Vadim Yu. Kaloshin

Nonlinearity 10 (1997), 1031-1046.

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We introduce a new potential-theoretic definition of the dimension spectrum $D_q$ of a probability measure for $q > 1$ and explain its relation to prior definitions. We apply this definition to prove that if $1 < q \leq 2$ and $\mu$ is a Borel probability measure with compact support in $\Bbb R^n$, then under almost every linear transformation from $\Bbb R^n$ to $\Bbb R^m$, the $q$-dimension of the image of $\mu$ is $\min(m,D_q(\mu))$; in particular, the $q$-dimension of $\mu$ is preserved provided $m \geq D_q(\mu)$. We also present results on the preservation of information dimension $D_1$ and pointwise dimension. Finally, for $0 \leq q < 1$ and $q > 2$ we give examples for which $D_q$ is not preserved by any linear transformation into $\Bbb R^m$. All results for typical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.

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