Global and local minima of $α$-Brjuno functions

TL;DR

The paper investigates global and local minima of the generalized Brjuno functions $B_\alpha$ for $\alpha\in(0,1]$. It extends a scaling analysis from the classical case $\alpha=1$, showing the global minimum of $B_1$ occurs at the golden conjugate $g$ and that its preimages under the Gauss map are local minima, then transfers these ideas to $\alpha<1$, proving a global minimum at $g$ for $\alpha\in(g,1)$ and a cusp-like minimum at $\gamma=\sqrt{2}-1$ when $\alpha=1/2$, with propagation via the $\alpha$-continued fraction dynamics. The authors establish lower semicontinuity for rational $\alpha$ and present a counterexample for certain irrational $\alpha$, along with an intermediate-value property and extensions to more general Brjuno variants. They also show monotonic and scaling phenomena that yield intervals in parameter space where the minimum is constant, and they discuss rich numerical evidence and open questions about existence, regularity, and behavior of $\alpha\mapsto \inf_x B_\alpha(x)$ across $[1/2,1]$. These results illuminate how arithmetical structure in the $\alpha$-continued fraction maps governs the shape and location of minima, with implications for small-divisor problems and rigidity phenomena related to Brjuno-type conditions.

Abstract

The main goal of this article is to analyze some peculiar features of the global (and local) minima of $α$-Brjuno functions $B_α$ where $α\in(0,1].$ Our starting point is the result by Balazard--Martin (2020), 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 $B_α$ for other values of $α$: we show that for $α\in (g,1)$ the global minimum is again attained at $g$, while for $α$ in a neighbourhood of $1/2$ the function $B_α$ attains its minimum at $γ:=\sqrt{2}-1$. The fact that the minimum of $B_α$ is attained when $α$ ranges a whole interval of parameters is non trivial. Indeed, we prove that $B_α$ is lower semicontinuous for all rational $α,$ but we also exhibit an irrational $α$ for which $B_α$ is not lower semicontinuous. %We also prove that if $α$ is rational then $B_α$ is lower semicontinuous. This property does not hold in general, in fact we show that $B_α$ is not lower semicontinuous for a suitable irrational $α.$

Figures (10)

• Figure 1: The graph of the Brjuno function $B_{1}$ associated to Gauss map.
• Figure 2: (Approximate) behaviour of the minimum point and minimum value as $\alpha$ ranges in $[0,1]$; shaded regions correspond to parameter values for which the numerical evidence is confirmed by our analytical results.
• Figure 3: The graph showing lower semicontinuity fails for some irrational $\hat{\alpha}$; here a=2 and $\hat{\alpha}=\frac{2+g}{3+g}.$
• Figure 4: Graph of $\left(\log|x-g|,\log (B_{1}(x)-B_{1}(g))\right)$ for various values of $x$ including $x_n=\Phi^{n}(1-g)$; the dots are represented in two different colours accordingly to the sign of $x-g$.
• Figure 5: The three curves that bound $B_1$ from below, seen at three different level of zoom around the point $(g^2,B_1(g^2))$.

Theorems & Definitions (53)

• Theorem 1.1: BaMa_20
• Definition 1.2
• Conjecture 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1: MaMoYo_97MoCaMa_99
• Lemma 2.2
• Lemma 2.3