Global and local minima of α-Brjuno functions

The aim of this article is to analyze some peculiar features of the global (and local) minima of α-Brjuno functions B_α where α∈[1/2,1]. Our starting point is the result by Balazard–Martin (2020), who showed that the minimum of B_1 is attained at g:=√(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∈ (g,1) the global minimum is again attained at g, while for α=1/2 the function B_1/2 attains its minimum at γ:=√(2)-1.