Dynamical self-similarity, $L^{q}$-dimensions and Furstenberg slicing in $\mathbb{R}^d$

TL;DR

This paper generalizes results on the $L^q$-dimensions of self-similar measures to higher dimensions, providing a simpler proof and establishing new slicing theorems related to Furstenberg's conjecture.

Contribution

It extends a theorem on $L^q$-dimensions to arbitrary dimensions and introduces a novel, simplified proof method, also deriving higher rank slicing theorems.

Findings

$L^q$-dimensions of self-similar measures in $\

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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.