Assouad spectrum of Gatzouras-Lalley carpets

TL;DR

This paper derives a formula for the Assouad spectrum of Gatzouras-Lalley carpets, revealing novel properties like differentiability and concavity, using advanced multifractal and optimization techniques.

Contribution

It provides the first explicit formula for the Assouad spectrum of Gatzouras-Lalley carpets, highlighting new spectral behaviors and introducing a versatile covering framework.

Findings

Assouad spectrum can be a non-trivial differentiable function on (0,1)

The spectrum can be strictly concave on open intervals

New properties of the Assouad spectrum for self-affine sets are established

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.