Rotation Domains and Solenoid Models in Higher-Dimensional Dynamics

Definitions and mathematical statement

This entry is a bundle of questions in complex dimension two. A rotation domain is understood in the report's dynamical setting, with a suitable iterate analytically conjugate to a rotation; a Siegel ball is the corresponding ball-type domain. The ambient map class, rotation rank, iterate, and exact ball convention are not reproduced. Retain the following distinct targets:

1. Does a Siegel ball exist whose boundary is homeomorphic to $S^3$? The requested boundary property is topological, with no smoothness asserted.
2. Does a rotation domain exist that is biholomorphic to $A\times{\mathbb{C}}$, where $A$ is an annulus? The annulus modulus and allowed map class are unspecified.
3. Describe the domain of convergence of the linearizing map for a two-dimensional rotation domain. Neither the chosen linearization nor a proposed geometric description is given.
4. For a complex Hénon map with connected Julia set $J$, when is the report's natural map $J^-\setminus K\to\Sigma$ to the complex solenoid one-to-one? When is $J$ a quotient of a solenoid, and which equivalence relations on that solenoid occur?
5. Find an effective structural characterization of connectedness of the Julia set for Hénon-like maps; the source gives no specific proposed criterion.

For orientation, for an invertible map $f:{\mathbb{C}}^2\to{\mathbb{C}}^2$ of Hénon type, customary notation is \[ K^\pm=\{z:\{f^{\pm n}(z):n\geq0\}\text{ is bounded}\},\qquad J^\pm=\partial K^\pm,\qquad K=K^+\cap K^-,\quad J=J^+\cap J^-. \] The report's conventions must be confirmed. In particular, the construction and domain of the "natural map," the precise complex solenoid $\Sigma$, its degree, and the allowed quotient equivalence relations are absent. A complex solenoid is not silently replaced by a compact real circle solenoid, and connectedness alone is not supplemented by hyperbolicity. The Hénon-like class in the last question is likewise not defined here. These omissions prevent turning the bundle into one self-contained theorem or assigning one verified status to all its parts.

Short English statement

Ask about sphere-boundary Siegel balls, annulus-times-plane rotation domains, linearization domains, solenoid descriptions for connected Hénon Julia sets, and connectedness criteria for Hénon-like maps. Several map and solenoid definitions are missing.

Sources

• Local UnsolvedMath registry, permanent ID $30001089$, source code catalogue entry, "Rotation Domains and Solenoid Models in Higher-Dimensional Dynamics." No individual proposer is supplied.
• Trends and Developments in Complex Dynamics (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/48, pp. 2703--2762: https://doi.org/10.4171/owr/2008/48. This is the sole supplied external reference.
• Supplied literature triage, checked 2026-08-21: status "open." It describes several proposed problems rather than one complete claim and reports no authoritative source establishing all of them. This is supplied dated curation, not independently verified; it does not individually certify every subproblem open, and time-sensitive status and cited claims require independent verification.