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 (Fund Math 218(3): 193–224, 2012). https://doi.org/10.4064/fm218-3-1, 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 over 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.
