Dimension of homogeneous iterated function systems with algebraic translations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929336150392832 |
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| author | Feng, De-Jun Feng, Zhou |
| author_facet | Feng, De-Jun Feng, Zhou |
| contents | Let $ μ$ be the self-similar measure associated with a homogeneous iterated function system $ Φ= \{ λx + t_j \}_{j=1}^m $ on ${\Bbb R}$ and a probability vector $ (p_{j})_{j=1}^m$, where $0\neq λ\in (-1,1)$ and $t_j\in {\Bbb R}$. Recently by modifying the arguments of Varjú (2019), Rapaport and Varjú (2024) showed that if $t_1,\ldots, t_m$ are rational numbers and $0<λ<1$, then $$ \dim μ=\min\Big \{ 1, \; \frac{\sum_{j=1}^m p_{j}\log p_{j}}{ \log |λ| }\Big\}$$ unless $ Φ$ has exact overlaps. In this paper, we further show that the above equality holds in the case when $t_1,\ldots, t_m$ are algebraic numbers and $0<|λ|<1$. This is done by adapting and extending the ideas employed in the recent papers of Breuillard, Rapaport and Varjú. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03124 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dimension of homogeneous iterated function systems with algebraic translations Feng, De-Jun Feng, Zhou Dynamical Systems Classical Analysis and ODEs 28A80, 42A85 Let $ μ$ be the self-similar measure associated with a homogeneous iterated function system $ Φ= \{ λx + t_j \}_{j=1}^m $ on ${\Bbb R}$ and a probability vector $ (p_{j})_{j=1}^m$, where $0\neq λ\in (-1,1)$ and $t_j\in {\Bbb R}$. Recently by modifying the arguments of Varjú (2019), Rapaport and Varjú (2024) showed that if $t_1,\ldots, t_m$ are rational numbers and $0<λ<1$, then $$ \dim μ=\min\Big \{ 1, \; \frac{\sum_{j=1}^m p_{j}\log p_{j}}{ \log |λ| }\Big\}$$ unless $ Φ$ has exact overlaps. In this paper, we further show that the above equality holds in the case when $t_1,\ldots, t_m$ are algebraic numbers and $0<|λ|<1$. This is done by adapting and extending the ideas employed in the recent papers of Breuillard, Rapaport and Varjú. |
| title | Dimension of homogeneous iterated function systems with algebraic translations |
| topic | Dynamical Systems Classical Analysis and ODEs 28A80, 42A85 |
| url | https://arxiv.org/abs/2405.03124 |