On the $L^q$-spectrum of planar self-affine measures

TL;DR

This paper analyzes the $L^q$-spectrum of planar self-affine measures supported on box-like sets, providing explicit formulas, differentiability results, and applications to various fractal dimensions, including cases with rotational and reflectional symmetries.

Contribution

It introduces a method to compute the $L^q$-spectrum for a broad class of self-affine measures, including those with rotations and reflections, and extends existing results to graph-directed cases.

Findings

Derived a closed-form $L^q$-spectrum expression under certain conditions.

Proved differentiability of the $L^q$-spectrum at $q=1$ in complex cases.

Extended spectral existence results to graph-directed self-similar measures.

Abstract

We study the dimension theory of a class of planar self-affine multifractal measures. These measures are the Bernoulli measures supported on box-like self-affine sets, introduced by the author, which are the attractors of iterated function systems consisting of contracting affine maps which take the unit square to rectangles with sides parallel to the axes. This class contains the self-affine measures recently considered by Feng and Wang as well as many other measures. In particular, we allow the defining maps to have non-trivial rotational and reflectional components. Assuming the rectangular open set condition, we compute the $L^q$-spectrum by means of a $q$-modified singular value function. A key application of our results is a \emph{closed form expression} for the $L^q$-spectrum in the case where there are no mappings that switch the coordinate axes. This is useful for computational purposes and also allows us to prove differentiability of the $L^q$-spectrum at $q=1$ in the more difficult `non-multiplicative' situation. This has applications concerning the Hausdorff, packing and entropy dimension of the measure as well as the Hausdorff and packing dimension of the support. Due to the possible inclusion of axis reversing maps, we are led to extend some results of Peres and Solomyak on the existence of $L^q$-spectrum of self-similar measures to the graph-directed case.