Global and local minima of $α$-Brjuno functions

Abstract

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 $α.$