PreprintCC BY-NC-SA 4.0arXiv
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 Lq spectrum, which we denote by ϑ(q). We prove a variational formula for ϑ, from which it follows that multifractal formalism holds at all values q for which ϑ(q) is differentiable. However, in general, ϑ 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.
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BibTeX
@misc{jordan2026multifractal,
title = {{Multifractal analysis of Gatzouras--Lalley measures}},
author = {Thomas Jordan and István Kolossváry and Alex Rutar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35092},
eprint = {2609.35092},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35092},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Multifractal analysis of Gatzouras--Lalley measures
AU - Thomas Jordan
AU - István Kolossváry
AU - Alex Rutar
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35092
UR - https://arxiv.org/abs/2609.35092
AB - 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 ϑ(q). We prove a variational formula for ϑ, from which it follows that multifractal formalism holds at all values q for which ϑ(q) is differentiable. However, in general, ϑ 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.
ER -