Assouad spectrum of Gatzouras-Lalley carpets

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.

arXiv comment43 pages, 6 figures; v2: A minor auxiliary lemma (Lemma 4.7) in v1 was incorrect as stated and it has been removed; all other results unchanged. Numbering is different from v1; v3 Typo fixes and other minor updates. To appear in Adv. Math; v4 Fixes numbering issue and missing figure components caused by arXiv compilation issues

References in corpus (10)

• Assouad dimension of self-affine carpets
• The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result
• Rigidity of quasisymmetric mappings on self-affine carpets
• Spherical maximal functions and fractal dimensions of dilation sets
• Intermediate dimensions of Bedford-McMullen carpets with applications to Lipschitz equivalence
• Non-uniqueness of ergodic measures with full Hausdorff dimension on a Gatzouras-Lalley carpet
• Attainable forms of Assouad spectra
• The $L^q$ spectrum of self-affine measures on sponges
• Assouad type dimensions of infinitely generated self-conformal sets
• Assouad-type Dimensions of Overlapping Self-affine Sets