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Dynamical self-similarity, $L^{q}$-dimensions and Furstenberg slicing in $\mathbb{R}^d$
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abstract
We extend a theorem of the second author on the $L^q$-dimensions of dynamically driven self-similar measures from the real line to arbitrary dimension. Our approach provides a novel, simpler proof even in the one-dimensional case. As consequences, we show that, under mild separation conditions, the $L^q$-dimensions of homogeneous self-similar measures in $\mathbb{R}^d$ take the expected values, and we derive higher rank slicing theorems in the spirit of Furstenberg's slicing conjecture.
Forward citations
Cited by 2 Pith papers
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Smooth projections of self-similar measures
A spectral-gap criterion gives Sobolev regularity for prescribed projections of self-similar measures and yields explicit examples such as singular measures with all line projections smooth.
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On the $L^q$ dimension of stationary measures for M\"{o}bius iterated function systems
For Möbius iterated function systems satisfying the strongly Diophantine condition, the Lq spectrum of the stationary measure either equals the expected value for all q, or it becomes a linear function with slope belo...
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