On absolute continuity of inhomogeneous and contracting on average self-similar measures
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915559585611776 |
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| author | Kittle, Samuel Kogler, Constantin |
| author_facet | Kittle, Samuel Kogler, Constantin |
| contents | We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In fact, for $d\geq 1$ and any given rotations in $O(d)$ acting irreducibly on $\mathbb{R}^d$ as well as any distinct translations, all having algebraic coefficients, we construct absolutely continuous self-similar measures with the given rotations and translations. We furthermore strengthen Varjú's result for Bernoulli convolutions, treat complex Bernoulli convolutions and in dimension $\geq 3$ improve the condition on absolute continuity by Lindenstrauss-Varjú. Moreover, self-similar measures of contracting on average measures are studied, which may include expanding similarities in their support. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On absolute continuity of inhomogeneous and contracting on average self-similar measures Kittle, Samuel Kogler, Constantin Dynamical Systems Classical Analysis and ODEs Probability We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In fact, for $d\geq 1$ and any given rotations in $O(d)$ acting irreducibly on $\mathbb{R}^d$ as well as any distinct translations, all having algebraic coefficients, we construct absolutely continuous self-similar measures with the given rotations and translations. We furthermore strengthen Varjú's result for Bernoulli convolutions, treat complex Bernoulli convolutions and in dimension $\geq 3$ improve the condition on absolute continuity by Lindenstrauss-Varjú. Moreover, self-similar measures of contracting on average measures are studied, which may include expanding similarities in their support. |
| title | On absolute continuity of inhomogeneous and contracting on average self-similar measures |
| topic | Dynamical Systems Classical Analysis and ODEs Probability |
| url | https://arxiv.org/abs/2409.18936 |