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

Part of: Classical measure theory

Published online by Cambridge University Press:  26 June 2019

ARIEL RAPAPORT
ARIEL RAPAPORT*
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, CambridgeCB3 0WA, UK email [email protected]
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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 $L^{q}(\mathbb{R})$.

Keywords

self-similar sets and measuresabsolute continuitydimension conservation

MSC classification

Primary: 28A80: Fractals
Secondary: 28A78: Hausdorff and packing measures
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 12 , December 2020 , pp. 3438 - 3456
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Copyright
© Cambridge University Press, 2019

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