Closing Lemma for Diffeomorphism (Dynamical Systems)

Definitions and mathematical statement

Fix an integer $r\ge2$ and a compact smooth manifold $M$ without boundary. Let $\operatorname{Diff}^r(M)$ be the space of bijective $C^r$ maps with $C^r$ inverses, equipped with the $C^r$ topology: maps and their derivatives through order $r$ must be uniformly close in charts. A point $p$ is nonwandering for $f$ if every open neighborhood $U$ of $p$ satisfies \[ f^n(U)\cap U\ne\varnothing \quad\text{for some integer }n\ge1. \] Write $\Omega(f)$ for this set. The closing conjecture asks whether \[ \forall f\ \forall p\in\Omega(f)\ \forall C^r\text{ neighborhoods }V\ni f, \quad\exists g\in V\ \exists m\ge1\quad g^m(p)=p. \] The source's undefined symbol $\omega_f$ is interpreted as the nonwandering set in the standard closing-lemma formulation. The point $p$ itself is to become periodic, and the period is not prescribed. There is also a $C^\infty$ version using its usual smooth topology.

For flows, the analogous question replaces $f$ by a $C^r$ vector field, uses its flow to define nonwandering points, and asks for arbitrarily small $C^r$ perturbations creating a periodic orbit through a given nonsingular nonwandering point. Equilibria must be treated separately. These higher-regularity targets retain the specified perturbation topology.

Short English statement

Can every nonwandering point be made periodic by an arbitrarily small perturbation controlling derivatives through any prescribed higher order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3432 (OPG-48767), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Closing Lemma for Diffeomorphism (Dynamical Systems). Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/closing_lemma_for_diffeomorphism_dynamical_systems.