Exact Hausdorff dimension of some sofic self-affine fractals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929619874086912 |
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| author | Alibabaei, Nima |
| author_facet | Alibabaei, Nima |
| contents | Previous work has shown that the Hausdorff dimension of sofic affine-invariant sets is expressed as a limit involving intricate matrix products. This limit has typically been regarded as incalculable. However, in several highly non-trivial cases, we demonstrate that the dimension can in fact be calculated explicitly. Specifically, the dimension is expressed as the solution to an infinite-degree equation with explicit coefficients, which also corresponds to the spectral radius of a certain linear operator. Our result provides the first non-trivial calculation of the exact Hausdorff dimension of sofic sets in $\mathbb{R}^3$. This is achieved by developing a new technique inspired by the work of Kenyon and Peres (1998). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05805 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Exact Hausdorff dimension of some sofic self-affine fractals Alibabaei, Nima Dynamical Systems 28A80, 28D20, 37B40 Previous work has shown that the Hausdorff dimension of sofic affine-invariant sets is expressed as a limit involving intricate matrix products. This limit has typically been regarded as incalculable. However, in several highly non-trivial cases, we demonstrate that the dimension can in fact be calculated explicitly. Specifically, the dimension is expressed as the solution to an infinite-degree equation with explicit coefficients, which also corresponds to the spectral radius of a certain linear operator. Our result provides the first non-trivial calculation of the exact Hausdorff dimension of sofic sets in $\mathbb{R}^3$. This is achieved by developing a new technique inspired by the work of Kenyon and Peres (1998). |
| title | Exact Hausdorff dimension of some sofic self-affine fractals |
| topic | Dynamical Systems 28A80, 28D20, 37B40 |
| url | https://arxiv.org/abs/2412.05805 |