Fractal and multifractal dimensions of prevalent measures
Lars Olsen
28A80multifractalsHausdorff dimensionpacking dimensionlocal dimensionprevalenceshyness
Let $K$ be a compact subset of $\mathbb{R}^{d}$ and write $\mathcal{P}(K)$ for the family of Borel probability measures on $K$. In this paper we study different fractal and multifractal dimensions of measures $\mu$ in $\mathcal{P}(K)$ that are generic in the sense of prevalence. We first prove a general result, namely, for an arbitrary ``dimension'' function $\Delta: \mathcal{P}(K) \to \mathbb{R}$ satisfying various natural scaling and monotonicity conditions, we obtain a formula for the ``dimension'' $\Delta(\mu)$ of a prevalent measure $\mu$ in $\mathcal{P}(K)$; this is the content of Theorem 1.1. By applying Theorem 1.1 to appropriate choices of $\Delta$ we obtain the following results: \begin{itemize}\item By letting $\Delta(\mu)$ equal the (lower or upper) local dimension of $\mu$ at a point $x \in K$ and applying Theorem 1.1 to this particular choice of $\Delta$, we compute the (lower and upper) local dimension of a prevalent measure $\mu$ in $\mathcal{P}(K)$. \item By letting $\Delta(\mu)$ equal the multifractal spectrum of $\mu$ and applying Theorem 1.1 to this particular choice of $\Delta$, we compute the multifractal spectrum of a prevalent measure $\mu$ in $\mathcal{P}(K)$. \item Finally, by letting $\Delta(\mu)$ equal the Hausdorff or packing dimension of $\mu$ and applying Theorem 1.1 to this particular choice of $\Delta$, we compute the Hausdorff and packing dimension of a prevalent measure $\mu$ in $\mathcal{P}(K)$.\end{itemize} Perhaps surprisingly, in all cases our results are very different from the corresponding results for measures that are generic in the sense of Baire category.
Indiana University Mathematics Journal
2010
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10.1512/iumj.2010.59.3991
10.1512/iumj.2010.59.3991
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Indiana Univ. Math. J. 59 (2010) 661 - 690
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