Global and local minima of $α$-Brjuno functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917885697327104 |
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| author | Bakhtawar, Ayreena Carminati, Carlo Marmi, Stefano |
| author_facet | Bakhtawar, Ayreena Carminati, Carlo Marmi, Stefano |
| contents | The main goal of this article is to analyze some peculiar features of the global (and local) minima of $α$-Brjuno functions $B_α$ where $α\in(0,1].$ Our starting point is the result by Balazard--Martin (2020), who showed that the minimum of $B_1$ is attained at $g:=\frac{\sqrt 5 -1}{2}$; analyzing the scaling properties of $B_1$ near $g$ we shall deduce that all preimages of $g$ under the Gauss map are also local minima for $B_1$. Next we consider the problem of characterizing global and local minima of $B_α$ for other values of $α$: we show that for $α\in (g,1)$ the global minimum is again attained at $g$, while for $α$ in a neighbourhood of $1/2$ the function $B_α$ attains its minimum at $γ:=\sqrt{2}-1$. The fact that the minimum of $B_α$ is attained when $α$ ranges a whole interval of parameters is non trivial. Indeed, we prove that $B_α$ is lower semicontinuous for all rational $α,$ but we also exhibit an irrational $α$ for which $B_α$ is not lower semicontinuous. %We also prove that if $α$ is rational then $B_α$ is lower semicontinuous. This property does not hold in general, in fact we show that $B_α$ is not lower semicontinuous for a suitable irrational $α.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_17679 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global and local minima of $α$-Brjuno functions Bakhtawar, Ayreena Carminati, Carlo Marmi, Stefano Dynamical Systems 37F50, 11J70 The main goal of this article is to analyze some peculiar features of the global (and local) minima of $α$-Brjuno functions $B_α$ where $α\in(0,1].$ Our starting point is the result by Balazard--Martin (2020), who showed that the minimum of $B_1$ is attained at $g:=\frac{\sqrt 5 -1}{2}$; analyzing the scaling properties of $B_1$ near $g$ we shall deduce that all preimages of $g$ under the Gauss map are also local minima for $B_1$. Next we consider the problem of characterizing global and local minima of $B_α$ for other values of $α$: we show that for $α\in (g,1)$ the global minimum is again attained at $g$, while for $α$ in a neighbourhood of $1/2$ the function $B_α$ attains its minimum at $γ:=\sqrt{2}-1$. The fact that the minimum of $B_α$ is attained when $α$ ranges a whole interval of parameters is non trivial. Indeed, we prove that $B_α$ is lower semicontinuous for all rational $α,$ but we also exhibit an irrational $α$ for which $B_α$ is not lower semicontinuous. %We also prove that if $α$ is rational then $B_α$ is lower semicontinuous. This property does not hold in general, in fact we show that $B_α$ is not lower semicontinuous for a suitable irrational $α.$ |
| title | Global and local minima of $α$-Brjuno functions |
| topic | Dynamical Systems 37F50, 11J70 |
| url | https://arxiv.org/abs/2401.17679 |