Invariant Sets of Simplex Piecewise Translations

Definitions and mathematical statement

Write $T$ for a piecewise translation associated with a simplex and $F_T$ for the object called its invariant fundamental set in the supplied record. These are names for objects whose construction is not supplied, rather than fully specified mathematical data. In the usual vector-space terminology, a piecewise translation is a map on a partitioned domain whose restriction to each partition element has the form $x\mapsto x+v$, with a translation vector depending on the element. The record does not specify the ambient space, dimension, simplex, partition, translation vectors, domain, or boundary conventions.

Requested problem. Characterize $F_T$ in the cases described as the obtuse and unrestricted-translation cases, and determine whether, throughout the intended class, \[ F_T\text{ is connected} \qquad\text{and}\qquad F_T\text{ is a possibly non-convex polytope}. \] The intended class and its quantifiers remain unspecified. In particular, the record does not define which simplices are obtuse, which restrictions on translations are being removed, whether the two cases overlap, or how $F_T$ is selected. The word invariant could mean forward invariance $T(F_T)\subseteq F_T$ or equality $T(F_T)=F_T$; the supplied record does not settle this. The meaning of fundamental, including any underlying action or covering property, is also absent. Finally, the phrase possibly non-convex polytope requires a definition, since a polytope is conventionally convex in many contexts. These omissions prevent a complete quantified formulation without additional source material.

Short English statement

For the unspecified simplex-associated piecewise translations, describe the invariant fundamental set in the obtuse and unrestricted-translation cases and decide whether it is always connected and polyhedral in the intended, possibly non-convex sense. The construction and admissible cases are missing, so this is a qualified statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001290$, title Invariant Sets of Simplex Piecewise Translations, source problem code catalogue entry, collection Oberwolfach Reports Open Problems. The registry marks the statement as exact and internally readable; this is supplied metadata, not independent verification against the report. No individual proposer is identified.
• Supplied source citation: Dynamische Systeme (2010), Oberwolfach Reports, DOI 10.4171/owr/2009/32, pp. 1787--1842. https://doi.org/10.4171/owr/2009/32. The registry separately records proposed year 2009. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "OWR report": https://doi.org/10.4171/owr/2009/32. Literature checked at: 2026-08-21. This is supplied dated curation, not independently verified. Its status label is open, and its assessment says that the passage describes work in progress toward Smale's conjecture and gives no completed theorem resolving the stated further-progress problem. The supplied curation requests independent verification of time-sensitive status and cited claims.