On normal numbers and self-similar measures
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866910566950371328 |
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| author | Algom, Amir Baker, Simon Shmerkin, Pablo |
| author_facet | Algom, Amir Baker, Simon Shmerkin, Pablo |
| contents | Let $\lbrace f_i(x)=s_i \cdot x+t_i \rbrace$ be a self-similar IFS on $\mathbb{R}$ and let $β>1$ be a Pisot number. We prove that if $\frac{\log |s_i|}{\log β}\notin \mathbb{Q}$ for some $i$ then for every $C^1$ diffeomorphism $g$ and every non-atomic self similar measure $μ$, the measure $gμ$ is supported on numbers that are normal in base $β$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_10082 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On normal numbers and self-similar measures Algom, Amir Baker, Simon Shmerkin, Pablo Dynamical Systems Classical Analysis and ODEs Number Theory Let $\lbrace f_i(x)=s_i \cdot x+t_i \rbrace$ be a self-similar IFS on $\mathbb{R}$ and let $β>1$ be a Pisot number. We prove that if $\frac{\log |s_i|}{\log β}\notin \mathbb{Q}$ for some $i$ then for every $C^1$ diffeomorphism $g$ and every non-atomic self similar measure $μ$, the measure $gμ$ is supported on numbers that are normal in base $β$. |
| title | On normal numbers and self-similar measures |
| topic | Dynamical Systems Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2111.10082 |