Local Discrete Holomorphic Dynamics — Open Problem OP2

Definitions and mathematical statement

The supplied record names a Brjuno function $B$ and a function $r$, but does not define either function, their normalization, or the domain on which their sum is intended. Consequently, the following states the extension question precisely in terms of these unresolved source data.

Let $D\subseteq[0,1)$ be the intended domain on which $B(\alpha)$ is finite and real and $0<r(\alpha)<\infty$. Set \[ F(\alpha)=\log r(\alpha)+B(\alpha),\qquad \alpha\in D. \] A $1/2$-Hölder continuous extension of $F$ to $[0,1)$ means a function $H:[0,1)\to{\mathbb{R}}$ and a finite constant $C\geq0$ such that \[ H(\alpha)=F(\alpha)\quad\text{for every }\alpha\in D, \qquad |H(x)-H(y)|\leq C|x-y|^{1/2} \quad\text{for every }x,y\in[0,1). \] Problem (Abate, Problem OP2, 2011). For the functions and domain intended in the source, does such an extension exist? The endpoint $1$ is excluded; the displayed estimate requires one constant for the entire interval $[0,1)$.

The supplied source describes $B$ as diverging on a dense $G_\delta$ set of irrational numbers and reports numerical and theoretical evidence suggesting improved regularity of $\log r+B$. These are supplied contextual claims, not hypotheses specifying $B$, $r$, or $D$. The source attributes the extension question to Marmi, Mattei and Yoccoz. Recovering the missing definitions and checking this attribution are necessary before this schematic statement can serve as a fully specified mathematical problem.

Short English statement

Does the source's function $\log r+B$ extend to a real function on $[0,1)$ satisfying a uniform Hölder estimate of exponent $1/2$? The supplied excerpt leaves $r$, the precise Brjuno function $B$, and their intended domain undefined, so the formulation remains conditional on those definitions.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $800784$, source problem code catalogue entry, and title "Local Discrete Holomorphic Dynamics --- Open Problem OP2." Its supplied status is open; this label has not been independently verified.
• Abate, Open Problems in Local Discrete Holomorphic Dynamics (2011), Problem OP2. Supplied source URL: https://arxiv.org/abs/1107.4876. The record reports access on 2026-10-16 and extraction from arxiv_latex; these are supplied provenance details, not a claim of independent source access.
• The excerpt cites the unresolved reference labels [Car] for numerical experiments and further references, and [MMY] for theoretical results. No bibliographic entries for these labels are supplied. The excerpt attributes the proposed question to Marmi, Mattei and Yoccoz; that wording is preserved without independent verification.