Taylor-coefficient regularity of a Siegel conjugacy

Definitions and mathematical statement

The source specifies a holomorphic germ by \[ P_\rho'(z)=\lambda(1-z)^\rho,\qquad P_\rho(0)=0, \] and a Siegel linearizing map $h$. In the usual conjugacy convention, \[ h(0)=0,\qquad h'(0)\ne0,\qquad P_\rho(h(\zeta))=h(\lambda\zeta). \] For noninteger $\rho$, the branch of $(1-z)^\rho$ must be specified; the germ with $\log(1)=0$ defines it near zero, but the intended global domain is not supplied. Near zero define the coefficients by \[ \frac{h'(\zeta)}{1-h(\zeta)}=\sum_{\nu=0}^{\infty}a_\nu\zeta^\nu. \] The problem asks for a rigorous approximation of these coefficients by those of a "simplified equation", at least for small $\rho$. Neither that equation nor the parameter range or error criterion is included in the excerpt.

Its specific numerical benchmark is, for $\theta=(\sqrt5-1)/2$ and $\rho=1$, a uniform estimate such as \[ \left|a_\nu-\frac23\right|<0.1\qquad\text{for every integer }\nu\ge0. \] The usual rotation convention would set $\lambda=e^{2\pi i\theta}$, but the source excerpt does not explicitly give this relation. With that convention $P_1(z)=\lambda(z-z^2/2)$.

The normalization of $h$ is indispensable and is absent. For example replacing $h(\zeta)$ by $h(c\zeta)$, where defined and $c\ne0$, changes the coefficient $a_\nu$ to $c^{\nu+1}a_\nu$. Thus the proposed numerical estimate has no invariant meaning without the disk coordinate, radius, scaling, and phase convention. Imposing $h'(0)=1$ would itself force $a_0=1$, incompatible with the stated benchmark if it includes $\nu=0$; no such normalization is silently inserted.

The research summary discusses related boundary-critical-point linearization work but supplies no exact proof or citation for this estimate. Its description of an "indifferent critical point" should not replace the actual derivative formula: in the usual Siegel setting zero is the indifferent fixed point, while for positive integer $\rho$ the displayed derivative vanishes at $z=1$. The record labels the exact statement open/unverified and reports no comprehensive modern resolution check.

Short English statement

Make the observed coefficient approximation for a Siegel linearization rigorous, including the proposed uniform closeness to $2/3$ in the golden-mean, $\rho=1$ case. The omitted simplified equation and especially the normalization of the linearizer must be recovered before the numerical bound is a well-defined claim.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $5300024$, source code catalogue entry; supplied statement, background, and research summary, updated 2026-08-06.
• Lennart Carleson (1992), problem 3, in Stony Brook, Open Problems in Dynamical Systems: https://www.math.stonybrook.edu/open-problems-dynamical-systems. Supplied source-TeX and PDF-text extraction accessed 2026-07-29; no comprehensive modern resolution check was available.
• The summary mentions related Siegel-linearization and "Herman-like" expansion work without an identifiable publication or the simplified equation.