On dissonance of self-conformal measures in $\mathbb{R}^d$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909911038820352 |
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| author | Pyörälä, Aleksi |
| author_facet | Pyörälä, Aleksi |
| contents | Let $μ$ be a self-conformal measure on $\mathbb{R}^d$. In this note we establish conditions for $μ$ under which $\dim(μ*ν) = \min\lbrace d,\dimμ+\dimν\rbrace$ holds when $ν$ is any Ahlfors-regular or self-conformal measure on $\mathbb{R}^d$. Our main result states the following sufficient condition: $μ$ is totally non-linear and not supported on a smooth hypersurface. We also establish sufficient (likely non-sharp) algebraic conditions for self-conformal measures which are not totally non-linear. The proofs combine recent results on scaling sceneries of self-conformal measures with a Marstrand-type projection theorem for product sets due to López and Moreira. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_14493 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On dissonance of self-conformal measures in $\mathbb{R}^d$ Pyörälä, Aleksi Dynamical Systems Classical Analysis and ODEs Primary 28A80, Secondary 37A10 Let $μ$ be a self-conformal measure on $\mathbb{R}^d$. In this note we establish conditions for $μ$ under which $\dim(μ*ν) = \min\lbrace d,\dimμ+\dimν\rbrace$ holds when $ν$ is any Ahlfors-regular or self-conformal measure on $\mathbb{R}^d$. Our main result states the following sufficient condition: $μ$ is totally non-linear and not supported on a smooth hypersurface. We also establish sufficient (likely non-sharp) algebraic conditions for self-conformal measures which are not totally non-linear. The proofs combine recent results on scaling sceneries of self-conformal measures with a Marstrand-type projection theorem for product sets due to López and Moreira. |
| title | On dissonance of self-conformal measures in $\mathbb{R}^d$ |
| topic | Dynamical Systems Classical Analysis and ODEs Primary 28A80, Secondary 37A10 |
| url | https://arxiv.org/abs/2511.14493 |