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On the minimum of $σ$-Brjuno functions

Ayreena Bakhtawar, Carlo Carminati, Stefano Marmi

Abstract

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

On the minimum of $σ$-Brjuno functions

Abstract

-Brjuno functions were introduced in \cite{MaMoYo_06} as an interesting variant of the classical Brjuno function, where one substitutes the singularity at with the power law divergence As in the classical case, is a locally unbounded, highly irregular lower semi continuous function; from semi continuity property it easily follows that admits a global minimum but to locate it is quite a challenging problem. We prove that for , the unique global minimum of is achieved at the fixed point . Furthermore, we prove that these minimizers are locally stable, showing that the point of minimum remains constant for in a neighborhood of . 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 varies.
Paper Structure (12 sections, 16 theorems, 72 equations, 4 figures)

This paper contains 12 sections, 16 theorems, 72 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definition and properties of $\sigma$-Brjuno functions.
  3. Continued fraction preliminaries.
  4. The $\sigma$-Brjuno function.
  5. Relation with Classical Diophantine Classes.
  6. A priori estimates.
  7. Proof of Proposition \ref{['Reg']}.
  8. Global minimum of $B_{\sigma}.$
  9. Proof of Proposition \ref{['MT']}.
  10. Proof of Theorem \ref{['MTV2']}
  11. Scaling properties of $B_{n}$ for $n\ge 2$.
  12. Scaling properties of $B_{2}.$

Key Result

Theorem 1.1

Let $\sigma=n \in \mathbb N.$ Then the $\sigma$-Brjuno function achieves its unique global minimum at the point that is

Figures (4)

  • Figure 1: (Approximate) behaviour of the minimum point and minimum value as $\sigma$ ranges. The lower part shows the minimum value compared to the plot of the function $\sigma \mapsto B_\sigma(\eta_n)$ for $1\leq n \leq 8$, where $\eta_n=(\sqrt{n^2+4}-n)/2=[0; \overline{n}]$ is the n-th fixed point of the Gauss map. The point of minimum seems to undergo a jump at values $\sigma^*_n$ defined by equation \ref{['sstar']}. The yellow vertical lines in the upper part are drawn at the value $\sigma=\sigma^{*}_{n}.$
  • Figure 2: The function $B_3$ and the lower bounds $g$ (in yellow) and $g_k$ (in red) for $k=1,2,3,4$.
  • Figure 3: The difference $\delta_n:= g(\xi_n)-B_n(\eta_{n+1})$ (on the $x$ axis) versus $n$ (on the $y$-axis) for $n\in [2,800]$. The graph is plotted in loglog scale, and the fact that for $n>>1$ these points appear to be aligned on a straight line of slope $-1/3$ derives from the asymptotic formula $\delta_n \sim \frac{1}{2}n^{-3}$ as $n\to +\infty$.
  • Figure 4: Graphs of $B_{\sigma}$ for different values of $\sigma$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Remark 1.4
  • proof : Proof of Theorem \ref{['MTV']}
  • Conjecture 1.5
  • Theorem 1.6
  • Proposition 2.1: Integrability
  • proof
  • Remark 2.2: Lower semi-continuity
  • ...and 22 more