Abstract
Typical projections of simple multifractal measures with generalized dimensionsD q onto subspaces of dimensionD are considered. It is known that forD o > D almost all projections have Euclidean support. Here it is shown that if in additionD∞ increases beyondD, a typical projection changes from a singular continuous distribution to an absolutely continuous measure with a squareintegrable, or even differentiable density, and thus from a multifractal to an ordinary distribution with trivial singularity spectrum. Since projections of strictly self-similar measures can be regarded as invariant distributions of iterated function systems, such a transition is found also there and is expected to occur in related systems.
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Radons, G. A new transition for projections of multifractal measures and random maps. J Stat Phys 72, 227–239 (1993). https://doi.org/10.1007/BF01048048
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DOI: https://doi.org/10.1007/BF01048048