Multifractal analysis of Gatzouras--Lalley measures

Abstract

We study the multifractal analysis of self-affine measures supported on planar carpets of type Gatzouras--Lalley. We work primarily with Olsen's Hausdorff-dimensional variant of the $L^q$ spectrum, which we denote by $\vartheta(q)$. We prove a variational formula for $\vartheta$, from which it follows that multifractal formalism holds at all values $q$ for which $\vartheta(q)$ is differentiable. However, in general, $\vartheta$ need not be differentiable and the multifractal formalism may fail. In fact, the multifractal spectrum may be non-differentiable and have jump discontinuities in the interior of its support, and there may be multiple phase transitions at both negative and positive values of $q$.