On the minimum of $\ddot{I}ƒ$-Brjuno functions

Abstract

$\ddot{I}ƒ$-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 $x^{-1/\ddot{I}ƒ},$ $(\ddot{I}ƒ>0).$ As in the classical case, $B_{\ddot{I}}ƒ$ is a locally unbounded, highly irregular lower semi continuous function; from semi continuity property it easily follows that $B_{\ddot{I}}ƒ$ admits a global minimum but to locate it is quite a challenging problem. We prove that for $\ddot{I}ƒ=n\in \mathbb{N}$, 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 $\ddot{I}ƒ$ 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 $\ddot{I}ƒ$ varies.

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