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On self-similar measures with absolutely continuous projections and dimension conservation in each direction
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 40 / Issue 12 / December 2020
- Published online by Cambridge University Press:
- 26 June 2019, pp. 3438-3456
- Print publication:
- December 2020
-
- Article
- Export citation
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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 
$L^{q}(\mathbb{R})$.
