On the minimum of $σ$-Brjuno functions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915846546259968 |
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| author | Bakhtawar, Ayreena Carminati, Carlo Marmi, Stefano |
| author_facet | Bakhtawar, Ayreena Carminati, Carlo Marmi, Stefano |
| contents | $σ$-Brjuno functions were introduced in \cite{MaMoYo_06} as an interesting variant of the classical Brjuno function, where one substitutes the $\log$ singularity at $x=0$ with the power law divergence $x^{-1/σ},$ $(σ>0).$ As in the classical case, $B_σ$ is a locally unbounded, highly irregular lower semi continuous function; from semi continuity property it easily follows that $B_σ$ admits a global minimum but to locate it is quite a challenging problem. We prove that for $σ=n \in \mathbb{N}$, the unique global minimum of $B_n$ is achieved at the fixed point $ [0; \overline{n+1}]$. Furthermore, we prove that these minimizers are locally stable, showing that the point of minimum remains constant for $σ$ in a neighborhood of $n$. Finally, we discuss the scaling behavior near these minima and we formulate a conjecture about the phase transitions for the location of the minimizer as $σ$ varies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08378 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the minimum of $σ$-Brjuno functions Bakhtawar, Ayreena Carminati, Carlo Marmi, Stefano Dynamical Systems $σ$-Brjuno functions were introduced in \cite{MaMoYo_06} as an interesting variant of the classical Brjuno function, where one substitutes the $\log$ singularity at $x=0$ with the power law divergence $x^{-1/σ},$ $(σ>0).$ As in the classical case, $B_σ$ is a locally unbounded, highly irregular lower semi continuous function; from semi continuity property it easily follows that $B_σ$ admits a global minimum but to locate it is quite a challenging problem. We prove that for $σ=n \in \mathbb{N}$, the unique global minimum of $B_n$ is achieved at the fixed point $ [0; \overline{n+1}]$. Furthermore, we prove that these minimizers are locally stable, showing that the point of minimum remains constant for $σ$ in a neighborhood of $n$. Finally, we discuss the scaling behavior near these minima and we formulate a conjecture about the phase transitions for the location of the minimizer as $σ$ varies. |
| title | On the minimum of $σ$-Brjuno functions |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2603.08378 |