The matrix potential game and structures of self-affine sets
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909738076209152 |
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| author | Howat, Richard A. Mitchell, Andrew Samuel, Tony |
| author_facet | Howat, Richard A. Mitchell, Andrew Samuel, Tony |
| contents | We present a new variant of the potential game and show that certain compact subsets of $\R^n$, including a large class of self-affine sets, are winning in our game. We prove that sets with sufficiently strong winning conditions are non-empty, provide a lower bound for their Hausdorff dimension, show that they have good intersection properties, and provide conditions under which, given $M \in \N$, they contain a homothetic copy of every set with at most $M$ elements. The applications of our game to self-affine sets are new and complement the recent work of Yavicoli et al (Math. Z. 2022 and Int. Math. Res. Not. IMRN 2023) for self-similar sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The matrix potential game and structures of self-affine sets Howat, Richard A. Mitchell, Andrew Samuel, Tony Dynamical Systems Metric Geometry We present a new variant of the potential game and show that certain compact subsets of $\R^n$, including a large class of self-affine sets, are winning in our game. We prove that sets with sufficiently strong winning conditions are non-empty, provide a lower bound for their Hausdorff dimension, show that they have good intersection properties, and provide conditions under which, given $M \in \N$, they contain a homothetic copy of every set with at most $M$ elements. The applications of our game to self-affine sets are new and complement the recent work of Yavicoli et al (Math. Z. 2022 and Int. Math. Res. Not. IMRN 2023) for self-similar sets. |
| title | The matrix potential game and structures of self-affine sets |
| topic | Dynamical Systems Metric Geometry |
| url | https://arxiv.org/abs/2508.11577 |