On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910129265311744 |
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| author | Feng, Zhou |
| author_facet | Feng, Zhou |
| contents | Let $ K $ be a compact subset of the $d$-torus invariant under an expanding diagonal endomorphism with $ s $ distinct eigenvalues. Suppose the symbolic coding of $K$ satisfies weak specification. When $ s \leq 2 $, we prove that the following three statements are equivalent: (A) the Hausdorff and box dimensions of $ K $ coincide; (B) with respect to some gauge function, the Hausdorff measure of $ K $ is positive and finite; (C) the Hausdorff dimension of the measure of maximal entropy on $ K $ attains the Hausdorff dimension of $ K $. When $ s \geq 3 $, we find some examples in which (A) does not hold but (C) holds, which is a new phenomenon not appearing in the planar cases. Through a different probabilistic approach, we establish the equivalence of (A) and (B) for Bedford-McMullen sponges. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03213 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets Feng, Zhou Dynamical Systems 28A80, 37D35 Let $ K $ be a compact subset of the $d$-torus invariant under an expanding diagonal endomorphism with $ s $ distinct eigenvalues. Suppose the symbolic coding of $K$ satisfies weak specification. When $ s \leq 2 $, we prove that the following three statements are equivalent: (A) the Hausdorff and box dimensions of $ K $ coincide; (B) with respect to some gauge function, the Hausdorff measure of $ K $ is positive and finite; (C) the Hausdorff dimension of the measure of maximal entropy on $ K $ attains the Hausdorff dimension of $ K $. When $ s \geq 3 $, we find some examples in which (A) does not hold but (C) holds, which is a new phenomenon not appearing in the planar cases. Through a different probabilistic approach, we establish the equivalence of (A) and (B) for Bedford-McMullen sponges. |
| title | On the coincidence of the Hausdorff and box dimensions for some affine-invariant sets |
| topic | Dynamical Systems 28A80, 37D35 |
| url | https://arxiv.org/abs/2405.03213 |