A mathematical preprint shows that the spectra used to describe some planar fractal measures can be pinned down exactly in certain settings, but can also develop gaps, jumps and other irregular features. The work focuses on self-affine measures supported on Gatzouras-Lalley carpets. Its main message is conditional: the multifractal formalism works under specific assumptions and at points where the relevant fine spectrum is differentiable, but the paper also gives examples where that relationship fails.
The document is an arXiv preprint, version v1, dated 28 September 2026. It is a theoretical analysis, not an empirical study. There is no sample of people, field measurements or experimental comparison group; the objects under examination are mathematical measures and the iterated systems that generate them.
A precise map for a difficult calculation
The paper examines two related ways of describing how mass is distributed across a fractal. The fine Lq-spectrum captures detailed scaling behaviour, while the multifractal spectrum records how large the sets are at particular local-dimension values. The analysis combines variational optimisation with concave duality, a way of relating one optimisation problem to another. It also uses entropy, Lyapunov exponents, approximate-square estimates and uniform-density bounds.
Under the paper's main assumptions, the fine spectrum has an exact variational formula. In plain terms, its value can be found by minimising a quantity over probability assignments to the system's maps, or equivalently over assignments to its columns. This provides a general description of the fine spectrum within the stated class of systems.
The same duality explains when the multifractal formalism is exact. At any point where the fine spectrum has a derivative, the multifractal spectrum equals the corresponding concave-conjugate prediction from the fine spectrum. In this setting, differentiability marks the points where the two descriptions agree.
Where the usual relationship breaks
The paper also constructs measures for which that clean picture fails. A fine spectrum can be non-differentiable, while the multifractal spectrum sits strictly below the prediction obtained from its concave conjugate at a positive local-dimension value. The multifractal spectrum can also have an interior jump, meaning that it changes abruptly within the range of values it supports. These are existence results, not estimates of how often such behaviour occurs.
The study gives a precise test for another possible mismatch. At a fixed value of the Lq parameter, the coarse and fine spectra agree if and only if two functions describing column-dependent behaviour are both constant. Equality therefore requires the relevant column functions to show no variation at that parameter value.
For systems with two columns, the authors obtain an exact formula for the multifractal spectrum itself. At a chosen local-dimension value, the spectrum is the largest dimension among all probability assignments whose associated local dimension equals that target. This exact formula is restricted to the stated two-column setting.
Special cases and constructed extremes
Some more orderly systems behave better. When the defining iterated function system is homogeneous, the fine spectrum is differentiable and the fine multifractal formalism holds. By contrast, a nonhomogeneous carpet with two homogeneous columns and unequal column-dimension parameters has a nonempty open region of probability assignments that produces an interior jump in the multifractal spectrum. The result shows that the irregularity is not confined to one isolated parameter choice, although it does not say how common it is.
A concrete example in the plane uses an iterated function system with three maps and combines all three exceptional behaviours identified in the paper: a discontinuous multifractal spectrum, a non-differentiable fine spectrum and failure of the multifractal formalism. A separate computer-search example reports two phase transitions, one for negative values of the Lq parameter and one for positive values, together with a jump discontinuity and a continuous but non-differentiable point.
The main setup assumes that the probability vector is fully supported and that the system has separation of principal projections. The paper's strongest formulas therefore depend on explicit structural conditions, and its examples do not establish that discontinuities, formalism gaps or multiple phase transitions are typical across all Gatzouras-Lalley systems.
The acknowledgements report support for the work from Hungarian, UK, Canadian and Finnish research programmes. They also disclose limited use of ChatGPT 5.6 Sol to write Julia code for the computer search behind the phase-transition example and to help with minor TikZ edits for a figure.
Paper data and sources
Original title: Multifractal analysis of Gatzouras--Lalley measures
Authors: Thomas Jordan, István Kolossváry, Alex Rutar
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-09-28
DOI: Not available
Original paper · Full text