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On self-similar measures with absolutely continuous projections and dimension conservation in each direction

Published online by Cambridge University Press:  26 June 2019

ARIEL RAPAPORT
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WA, UK email ariel.rapaport@mail.huji.ac.il
Corresponding

Abstract

Relying on results due to Shmerkin and Solomyak, we show that outside a zero-dimensional set of parameters, for every planar homogeneous self-similar measure $\unicode[STIX]{x1D708}$ , with strong separation, dense rotations and dimension greater than $1$ , there exists $q>1$ such that $\{P_{z}\unicode[STIX]{x1D708}\}_{z\in S}\subset L^{q}(\mathbb{R})$ . Here $S$ is the unit circle and $P_{z}w=\langle z,w\rangle$ for $w\in \mathbb{R}^{2}$ . We then study such measures. For instance, we show that $\unicode[STIX]{x1D708}$ is dimension conserving in each direction and that the map $z\rightarrow P_{z}\unicode[STIX]{x1D708}$ is continuous with respect to the weak topology of $L^{q}(\mathbb{R})$ .

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Original Article
Copyright
© Cambridge University Press, 2019

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