The Brjuno and Wilton Functions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910869802188800 |
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| author | Burrin, Claire Lee, Seul Bee Marmi, Stefano |
| author_facet | Burrin, Claire Lee, Seul Bee Marmi, Stefano |
| contents | The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function $B(x)$ is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function $W(x)$ stems from the study of divisor sums and self-correlation functions in analytic number theory. We show that these perspectives are unified by the semi-Brjuno function $B_0(x)$. Namely, $B(x)$ and $W(x)$ can be expressed in terms of the even and odd parts of $B_0(x)$, respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions $Δ^+(x) = B^+(x) - 2B_0^+(x)$ and $Δ^-(x) = W^-(x) - 2B_0^-(x)$, the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Brjuno and Wilton Functions Burrin, Claire Lee, Seul Bee Marmi, Stefano Dynamical Systems 37F50, 11J70 The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function $B(x)$ is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function $W(x)$ stems from the study of divisor sums and self-correlation functions in analytic number theory. We show that these perspectives are unified by the semi-Brjuno function $B_0(x)$. Namely, $B(x)$ and $W(x)$ can be expressed in terms of the even and odd parts of $B_0(x)$, respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions $Δ^+(x) = B^+(x) - 2B_0^+(x)$ and $Δ^-(x) = W^-(x) - 2B_0^-(x)$, the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function. |
| title | The Brjuno and Wilton Functions |
| topic | Dynamical Systems 37F50, 11J70 |
| url | https://arxiv.org/abs/2503.08206 |