Foliations and Diffeomorphism Groups — Question 38

Definitions and mathematical statement

Let $I=\{a,b,c,d\}$. The supplied question considers four local $C^r$ diffeomorphisms $f_i$ of ${\mathbb{R}}$, each defined on a neighborhood of $[-1,1]$, such that \[ f_i(0)=0\qquad(i\in I). \] For a map $f$, define \[ \operatorname{Fix}(f)=\{x\in\operatorname{dom}(f):f(x)=x\}. \] The regularity parameter $r$ is not specified in the supplied record. Write \[ h=[f_a,f_b]\circ[f_c,f_d]. \] Here brackets denote commutators, but the source does not specify a convention. For example, one convention is $[f,g]=f\circ g\circ f^{-1}\circ g^{-1}$. Local inverses and these compositions are defined sufficiently near the common fixed point; an assertion on a larger domain requires additional domain specifications.

The source asks whether there exist local $C^r$ diffeomorphisms $\tilde f_i$, $C^r$-close to the corresponding $f_i$, that coincide with them outside $[-1,1]$, and satisfy the following two displayed conditions, transcribed without repair: \[ h=[\tilde f_a\tilde f_b][\tilde f_c\tilde f_d], \] \[ \operatorname{Fix}(f_a)\cap\operatorname{Fix}(f_b)\cap \operatorname{Fix}(f_c)\cap\operatorname{Fix}(f_d)\cap(-1,1)=\varnothing. \] These conditions do not specify a coherent perturbation problem as written. Each bracket in the first condition lacks a separator between two commutator arguments. The second condition refers to the original maps, whose fixed-point sets all contain $0\in(-1,1)$.

A possible intended repair is preservation of the product of commutators together with elimination of common fixed points of the perturbed maps: \[ [\tilde f_a,\tilde f_b]\circ[\tilde f_c,\tilde f_d]=h, \qquad \bigcap_{i\in I}\operatorname{Fix}(\tilde f_i)\cap(-1,1)=\varnothing. \] This repair is an unverified interpretation, not an established transcription of the question. A precise formulation also requires the allowed values of $r$, the topology and quantifier intended by $C^r$-closeness, the domains on which the maps must agree outside $[-1,1]$, and the domain on which the commutator identity must hold. In particular, equality of germs at $0$ and equality on a specified neighborhood are distinct requirements.

Short English statement

The supplied question asks whether four local diffeomorphisms of ${\mathbb{R}}$ with a common fixed point can be perturbed inside $[-1,1]$, preserving their product of commutators and removing common fixed points. This interpretation requires unverified corrections to both displayed target conditions. The literal fixed-point condition concerns the original maps and conflicts with their assumed common fixed point. Regularity, closeness, and composition domains also require clarification.

Sources

• Local UnsolvedMath registry: permanent ID $800097$, exact source problem code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 38." The supplied registry attributes the problem to Nariman and Yazdi, with proposed year 2025, and labels it open. This label is supplied curation, not an independently verified mathematical or literature assessment.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 38. Supplied source URL: https://arxiv.org/abs/2503.18274. The record reports access on 2026-10-16 and extraction by arxiv_latex; these provenance details have not been independently verified here.
• Supplied upstream record: ID $800097$, revision d371ee917ac9647fbb06cd6fc529deec6f3dda82; registry update timestamp 2026-10-16T14:00:00Z.