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

The main goal of this article is to analyze some peculiar features of the global (and local) minima of α-Brjuno functions Bα where α∈(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 B1 is attained at g:=5-12; analyzing the scaling properties of B1 near g we shall deduce that all preimages of g under the Gauss map are also local minima for B1. Next we consider the problem of characterizing global and local minima of Bα for other values of α: we show that for α∈(g,1) the global minimum is again attained at g, while for α in a neighbourhood of 1/2 the function Bα attains its minimum at γ:=2-1. The fact that the minimum of Bα is attained when α ranges over a whole interval of parameters is non trivial. Indeed, we prove that Bα is lower semicontinuous for all rational α, but we also exhibit an irrational α for which Bα is not lower semicontinuous.