Henon Maps: Open Problems — Question 5

Definitions and mathematical statement

Equip ${\mathbb{R}}^2$ with the standard symplectic form $\omega={\,\mathrm{d}} x\wedge{\,\mathrm{d}} y$. An entire symplectomorphism, in the sense explicitly supplied by the record, is a real analytic diffeomorphism $f:{\mathbb{R}}^2\to{\mathbb{R}}^2$ satisfying $f^*\omega=\omega$ for which there exists a holomorphic map $F:{\mathbb{C}}^2\to{\mathbb{C}}^2$ with $F|_{{\mathbb{R}}^2}=f$. The record does not require $F$ to be a holomorphic automorphism.

The source asks whether such an $f$ can have a nontrivial annular rotation domain, declaring a domain trivial when the whole dynamics is conjugate to a rotation. It does not specify the definition of an annular rotation domain, the regularity of either conjugacy, or any irrationality requirement.

A conditional formulation makes these missing choices explicit. Put $\mathbb T={\mathbb{R}}/{\mathbb{Z}}$, and define \[ T_\theta(t,s)=(t+\theta,s)\quad\text{on }\mathbb T\times(0,1), \qquad R_\varphi(x,y)=\left(x\cos(2\pi\varphi)-y\sin(2\pi\varphi),\ x\sin(2\pi\varphi)+y\cos(2\pi\varphi)\right). \] Fix classes $\mathscr C_{\mathrm{loc}}$ and $\mathscr C_{\mathrm{glob}}$ of admissible conjugacies, and a set $\Theta\subseteq\mathbb T$ of admissible local rotation angles. Under the interpretation that an annular rotation domain is an invariant open annulus conjugate to $T_\theta$, and that whole dynamics means dynamics on ${\mathbb{R}}^2$, the question is whether there exist an entire symplectomorphism $f$, an open set $U\subseteq{\mathbb{R}}^2$, an angle $\theta\in\Theta$, and a conjugacy $h:U\to\mathbb T\times(0,1)$ such that \[ f(U)=U,\qquad h\in\mathscr C_{\mathrm{loc}},\qquad h\circ f|_U=T_\theta\circ h, \] while \[ \nexists\,\varphi\in\mathbb T\ \nexists\,H:{\mathbb{R}}^2\to{\mathbb{R}}^2\text{ in }\mathscr C_{\mathrm{glob}} \quad H\circ f=R_\varphi\circ H. \] This is a conditional schema, not a uniquely determined formalization: the supplied record does not determine these conjugacy classes, the angle restriction, whether the rotation domain must be maximal, or whether whole dynamics instead refers to the complex extension.

Short English statement

Does a real analytic symplectomorphism of ${\mathbb{R}}^2$, extending holomorphically to ${\mathbb{C}}^2$, possess an annular region with rotation dynamics although its whole dynamics is not conjugate to a rotation? The source leaves the rotation-domain and conjugacy conventions unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $801053$, source problem code catalogue entry, title Henon Maps: Open Problems --- Question 5. The supplied registry labels the problem open; this status has not been independently verified.
• Berger and Bedford, Henon Maps: A List of Open Problems (2023), Question 5, https://arxiv.org/abs/2312.03907. The record attributes the question to Berger and Bedford and gives proposed year 2023. Its provenance reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of source access during this preparation.