Assouad spectrum of Gatzouras-Lalley carpets
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917105657446400 |
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| author | Banaji, Amlan Fraser, Jonathan M. Kolossváry, István Rutar, Alex |
| author_facet | Banaji, Amlan Fraser, Jonathan M. Kolossváry, István Rutar, Alex |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07168 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Assouad spectrum of Gatzouras-Lalley carpets Banaji, Amlan Fraser, Jonathan M. Kolossváry, István Rutar, Alex Dynamical Systems Classical Analysis and ODEs Metric Geometry 28A80 (Primary) 37C45, 49N15 (Secondary) 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. |
| title | Assouad spectrum of Gatzouras-Lalley carpets |
| topic | Dynamical Systems Classical Analysis and ODEs Metric Geometry 28A80 (Primary) 37C45, 49N15 (Secondary) |
| url | https://arxiv.org/abs/2401.07168 |