The main goal of my talk 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 the golden number $g$.

We shall refine this result in two directions: we consider the problem of characterizing local minima of $B_1$ and we consider the problem of characterizing global and local minima of $B_\alpha$ for other values of $\alpha$.