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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.05240 |
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| _version_ | 1866911891356385280 |
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| author | Pyörälä, Aleksi |
| author_facet | Pyörälä, Aleksi |
| contents | We show that if $\lbrace φ_i\rbrace_{i\in Γ}$ and $\lbrace ψ_j\rbrace_{j\inΛ}$ are self-affine iterated function systems on the plane that satisfy strong separation, domination and irreducibility, then for any associated self-affine measures $μ$ and $ν$, the inequality $$\dim_{\rm H}(μ*ν) < \min \lbrace 2, \dim_{\rm H} μ+ \dim_{\rm H} ν\rbrace$$ implies that there is algebraic resonance between the eigenvalues of the linear parts of $φ_i$ and $ψ_j$. This extends to planar non-conformal setting the existing analogous results for self-conformal measures on the line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_05240 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Resonance between planar self-affine measures Pyörälä, Aleksi Classical Analysis and ODEs Dynamical Systems 28A80, 37A10 We show that if $\lbrace φ_i\rbrace_{i\in Γ}$ and $\lbrace ψ_j\rbrace_{j\inΛ}$ are self-affine iterated function systems on the plane that satisfy strong separation, domination and irreducibility, then for any associated self-affine measures $μ$ and $ν$, the inequality $$\dim_{\rm H}(μ*ν) < \min \lbrace 2, \dim_{\rm H} μ+ \dim_{\rm H} ν\rbrace$$ implies that there is algebraic resonance between the eigenvalues of the linear parts of $φ_i$ and $ψ_j$. This extends to planar non-conformal setting the existing analogous results for self-conformal measures on the line. |
| title | Resonance between planar self-affine measures |
| topic | Classical Analysis and ODEs Dynamical Systems 28A80, 37A10 |
| url | https://arxiv.org/abs/2302.05240 |