On the minimum of $\sigma$-Brjuno functions

$\sigma$-Brjuno functions were introduced in \cite{MaMoYo_06} as an interesting variant of the classical Brjuno function, where one substitutes the $\log$ singularity at $x=0$ with the power law divergence $x^{-1/\sigma},$ $(\sigma>0).$ As in the classical case, $B_{\sigma}$ is a locally unbounded, highly irregular lower semi continuous function; from semi continuity property it easily follows that $B_{\sigma}$ admits a global minimum but to locate it is quite a challenging problem. We prove that for $\sigma=n \in \mathbb{N}$, the unique global minimum of $B_n$ is achieved at the fixed point $[0; \overline{n+1}]$. Furthermore, we prove that these minimizers are locally stable, showing that the point of minimum remains constant for $\sigma$ in a neighborhood of $n$. Finally, we discuss the scaling behavior near these minima and we formulate a conjecture about the phase transitions for the location of the minimizer as $\sigma$ varies.