Dimension of diagonal self-affine measures with exponentially separated projections
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917920712425472 |
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| author | Feng, Zhou |
| author_facet | Feng, Zhou |
| contents | Let $ μ$ be a self-affine measure associated with a diagonal affine iterated function system (IFS) $ Φ= \{ (x_{1}, \ldots, x_{d}) \mapsto ( r_{i, 1}x_{1} + t_{i,1}, \ldots, r_{i,d}x_{d} + t_{i,d}) \}_{i\inΛ} $ on $ \mathbb{R}^{d} $ and a probability vector $ p = (p_{i})_{i\inΛ}$. For $ 1 \leq j \leq d $, denote the $ j $-th the Lyapunov exponent by $ χ_{j} := \sum_{i\inΛ} - p_{i} \log | r_{i,j} |$, and define the IFS induced by $ Φ$ on the $j$-th coordinate as $ Φ_{j} := \{ x \mapsto r_{i,j}x + t_{i,j}\}_{i\inΛ}$. We prove that if $ χ_{j_{1}} \neq χ_{j_{2}} $ for $ 1 \leq j_{1} < j_{2} \leq d $, and $ Φ_{j}$ is exponentially separated for $ 1 \leq j \leq d $, then the dimension of $ μ$ is the minimum of $ d $ and its Lyapunov dimension. This confirms a conjecture of Rapaport by removing the additional assumption that the linear parts of the maps in $ Φ$ are contained in a 1-dimensional subgroup. One of the main ingredients of the proof involves disintegrating $ μ$ into random measures with convolution structure. In the course of the proof, we establish new results on dimension and entropy increase for these random measures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17378 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dimension of diagonal self-affine measures with exponentially separated projections Feng, Zhou Dynamical Systems 28A80, 37C45 Let $ μ$ be a self-affine measure associated with a diagonal affine iterated function system (IFS) $ Φ= \{ (x_{1}, \ldots, x_{d}) \mapsto ( r_{i, 1}x_{1} + t_{i,1}, \ldots, r_{i,d}x_{d} + t_{i,d}) \}_{i\inΛ} $ on $ \mathbb{R}^{d} $ and a probability vector $ p = (p_{i})_{i\inΛ}$. For $ 1 \leq j \leq d $, denote the $ j $-th the Lyapunov exponent by $ χ_{j} := \sum_{i\inΛ} - p_{i} \log | r_{i,j} |$, and define the IFS induced by $ Φ$ on the $j$-th coordinate as $ Φ_{j} := \{ x \mapsto r_{i,j}x + t_{i,j}\}_{i\inΛ}$. We prove that if $ χ_{j_{1}} \neq χ_{j_{2}} $ for $ 1 \leq j_{1} < j_{2} \leq d $, and $ Φ_{j}$ is exponentially separated for $ 1 \leq j \leq d $, then the dimension of $ μ$ is the minimum of $ d $ and its Lyapunov dimension. This confirms a conjecture of Rapaport by removing the additional assumption that the linear parts of the maps in $ Φ$ are contained in a 1-dimensional subgroup. One of the main ingredients of the proof involves disintegrating $ μ$ into random measures with convolution structure. In the course of the proof, we establish new results on dimension and entropy increase for these random measures. |
| title | Dimension of diagonal self-affine measures with exponentially separated projections |
| topic | Dynamical Systems 28A80, 37C45 |
| url | https://arxiv.org/abs/2501.17378 |