Movable Invariant Neighborhoods of Compact Flow Sets

Definitions and mathematical statement

A continuous flow on ${\mathbb{R}}^3$ is a continuous map $\varphi:{\mathbb{R}}\times{\mathbb{R}}^3\to{\mathbb{R}}^3$ such that $\varphi(0,x)=x$ and $\varphi(s+t,x)=\varphi(s,\varphi(t,x))$ for every $s,t\in{\mathbb{R}}$ and $x\in{\mathbb{R}}^3$. Write $\varphi_t(x)=\varphi(t,x)$. A set $B\subseteq{\mathbb{R}}^3$ is invariant if $\varphi_t(B)=B$ for every $t\in{\mathbb{R}}$.

For this draft, interpret a compact set $K\subseteq{\mathbb{R}}^3$ as movable in the usual shape-theoretic neighborhood sense: for every open neighborhood $O$ of $K$, there exists an open neighborhood $V$ of $K$, with $V\subseteq O$, such that for every open neighborhood $W$ of $K$ satisfying $W\subseteq V$, there is a continuous homotopy \[ H:V\times[0,1]\longrightarrow O, \qquad H(x,0)=x\quad(x\in V), \qquad H(V,1)\subseteq W. \] This homotopy need not fix $K$ pointwise and is not required to arise from the flow. The supplied record does not define movability or specify the regularity of the flow; the shape-theoretic interpretation and the continuous-flow convention above require confirmation against the cited source.

The question is whether, for every such flow $\varphi$, every compact invariant set $A\subseteq{\mathbb{R}}^3$, and every open set $U\subseteq{\mathbb{R}}^3$ with $A\subseteq U$, there exists a compact set $K\subseteq{\mathbb{R}}^3$ such that \[ A\subseteq K\subseteq U,\qquad \varphi_t(K)=K\quad\text{for every }t\in{\mathbb{R}}, \qquad K\text{ is movable}. \] Considering open neighborhoods suffices to express the question for arbitrary neighborhoods. The inclusions are not required to be strict, and the source does not require $K$ itself to be a neighborhood of $A$.

Short English statement

Must every neighborhood of a compact invariant set of a flow on ${\mathbb{R}}^3$ contain a movable compact invariant superset of that set? Here movability is interpreted in the standard shape-theoretic sense, subject to source confirmation. The supplied record labels the question open; this draft does not independently verify that status.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003527$, source problem code catalogue entry, title "Movable Invariant Neighborhoods of Compact Flow Sets." The supplied collection is Oberwolfach Reports Open Problems. No individual proposer is identified; the proposed year is recorded as 2017.
• Supplied primary provenance: Dynamische Systeme (2018), Oberwolfach Reports, pp. 1987--2046, DOI 10.4171/owr/2017/32: https://doi.org/10.4171/owr/2017/32. The background describes Oberwolfach Reports as reports of mathematical workshops and their open-problem discussions. The bibliography also labels this item "Oberwolfach Report 32/2017." The supplied year labels are retained without reconciliation.
• Supplied literature reference: Kuperberg, Open problems in dynamics, https://topology.nipissingu.ca/tp/reprints/v31/tp31132.pdf.
• Supplied literature reference: Sindelarova, Movable approximations, https://auetd.auburn.edu/bitstream/handle/10415/1316/SINDELAROVA_PETRA_14.pdf.
• Supplied literature reference: Oberwolfach Report 32/2017, https://doi.org/10.4171/owr/2017/32.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open for arbitrary compact invariant sets and states that positive results are restricted to special solenoids and constructed examples. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review. These are supplied editorial assertions, not mathematical verification performed for this draft.