Assouad spectrum of Gatzouras-Lalley carpets

TL;DR

This work computes the Assouad spectrum for a broad class of self-affine fractals, Gatzouras–Lalley carpets, by deriving an explicit formula: $ ext{dim}^{ heta}_{ m A}K= ext{dim}_{ m B}igl(oldsymbol{eta}igr)+rac{ au^{*}(oldsymbol{ heta})}{oldsymbol{ heta}}$, where $ au$ is the column pressure formed from the column-structure data and $ au^{*}$ is its concave conjugate, with a nonlinear parameter change $oldsymbol{ heta}( heta)$. The authors develop a variational framework based on the method of types and covering arguments using approximate squares to express $ ext{dim}^{ heta}_{ m A}K$ as a constrained maximisation over Bernoulli measures, then solve it via Lagrange duality and boundary analysis. The main contributions reveal novel behavior: the spectrum can be a nontrivial differentiable function on $(0,1)$, exhibit strict concavity on intervals, and show arbitrarily many phase transitions of odd order; these features arise from the inhomogeneous column structure and are explained through a detailed spectrum partition and variational solution. The results provide a powerful framework for coarse coverings in self-affine geometry and yield insights with potential extensions to broader self-affine sets, the lower spectrum, and intermediate dimensions. The techniques bridge multifractal analysis, large deviations (method of types), and convex duality to deliver an explicit, broadly applicable tool for analyzing local scaling in complex fractals.

Abstract

We study the fine local scaling properties of a class of self-affine fractal sets called Gatzouras-Lalley carpets. More precisely, we establish a formula for the Assouad spectrum of all Gatzouras-Lalley carpets as the concave conjugate of an explicit piecewise-analytic function combined with a simple parameter change. Our formula implies a number of novel properties for the Assouad spectrum not previously observed for dynamically invariant sets; in particular, the Assouad spectrum can be a non-trivial differentiable function on the entire domain $(0,1)$ and can be strictly concave on open intervals. Our proof introduces a general framework for covering arguments using techniques developed in the context of multifractal analysis, including the method of types from large deviations theory and Lagrange duality from optimisation theory.

Figures (7)

• Figure 1: Maps
• Figure 2: Attractor
• Figure 4: A depiction of the spectrum partition. The dotted lines are tangents to the function $g=\min\{\psi_{\underline{j}_1},\psi_{\underline{j}_2}\}$ at the points $t_{\min}$, $t_1$, and $t_{\max}$ corresponding to the left and right derivatives, where appropriate. The labels indicate the slopes of the dotted lines.
• Figure 5: A depiction of decomposition provided by \ref{['ic:piecewise']}. The coloured curves are of the form $\mathop{\mathrm{dim_B}}\nolimits\eta(K)+g_i^*(\phi(\theta))/\phi(\theta)$ for $i=1,2$. The spectrum is differentiable but not twice differentiable at each $\theta_{i,\min}$ and $\theta_{i,\max}$ for $i=1,2$.
• Figure 6: Plot of the Assouad spectrum corresponding to a system with 4 homogeneous columns, 3 of which are non-trivial. The function $g(t)$ is a minimum of affine lines, so the corresponding spectrum $\operatorname{dim}^{\theta}_{\mathrm{A}} K$ is a piecewise convex function. The slope of $g_i(t)$ corresponds to the value $\theta_i$, for $i=1,2,3$. The dotted lines correspond to the concave conjugates at each $t_i$ for $i=0,\ldots,3$, extended beyond the range given by the corresponding affine lines.

Theorems & Definitions (45)

• Definition 1.1
• Theorem 1
• Definition 2.1
• Lemma 2.2
• Corollary 2
• Proof 1
• Corollary 3
• Proof 2
• Corollary 4
• Proof 3