Global and local minima of $\alpha$-Brjuno functions

Abstract

The main goal of this article is to analyze some peculiar features of the global (and local) minima of $\alpha$-Brjuno functions $B_\alpha$ where $\alpha\in(0,1].$ 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 $B_\alpha$ for other values of $\alpha$: we show that for $\alpha\in (g,1)$ the global minimum is again attained at $g$, while for $\alpha$ in a neighbourhood of $1/2$ the function $B_{\alpha}$ attains its minimum at $\gamma:=\sqrt{2}-1$. The fact that the minimum of $B_\alpha$ is attained when $\alpha$ ranges a whole interval of parameters is non trivial. Indeed, we prove that $B_{\alpha}$ is lower semicontinuous for all rational $\alpha,$ but we also exhibit an irrational $\alpha$ for which $B_{\alpha}$ is not lower semicontinuous. %We also prove that if $\alpha$ is rational then $B_{\alpha}$ is lower semicontinuous. This property does not hold in general, in fact we show that $B_{\alpha}$ is not lower semicontinuous for a suitable irrational $\alpha.$