Math Conjectures

CrC^r closing lemma

OPENLandmarkConjectureProposed c. 1960 · Standard version

Canonical statement

Let MM be a closed smooth manifold, r≥2r\ge2, f∈Diff⁡r(M)f\in\operatorname{Diff}^{r}(M), and x∈Mx\in M a nonwandering point of ff: every neighborhood U∋xU\ni x has fn(U)∩U≠∅f^n(U)\cap U\ne\varnothing for some n>0n>0. For every CrC^r neighborhood U\mathcal U of ff, there is g∈Ug\in\mathcal U for which gm(x)=xg^m(x)=x for some m>0m>0.
View source LaTeX
Let \(M\) be a closed smooth manifold, \(r\ge2\), \(f\in\operatorname{Diff}^{r}(M)\), and \(x\in M\) a nonwandering point of \(f\): every neighborhood \(U\ni x\) has \(f^n(U)\cap U\ne\varnothing\) for some \(n>0\). For every \(C^r\) neighborhood \(\mathcal U\) of \(f\), there is \(g\in\mathcal U\) for which \(g^m(x)=x\) for some \(m>0\).
DYN-002:[Pugh1967ClosingLemma][PughRobinson1983ClosingLemma][BonattiCrovisier2016RecurrenceGenericity][Gao2022ClosingClaim]
Dynamical systems

Notes

The CrC^r closing lemma asks whether recurrence can always be converted into genuine periodicity by a small smooth perturbation: given a diffeomorphism ff of a compact manifold and a nonwandering point xx — one whose neighborhoods return to meet themselves under iteration — is there gg arbitrarily CrC^r-close to ff for which xx is periodic? The question took shape around 1960 within the closing-lemma program, and for r≥2r\ge2 it remains a basic open conjecture of smooth dynamics.

Pugh established the C1C^1 case [Pugh1967ClosingLemma], and with Robinson extended it to conservative and Hamiltonian systems [PughRobinson1983ClosingLemma]; these perturbation techniques underpin a broad C1C^1-generic theory of recurrence [BonattiCrovisier2016RecurrenceGenericity]. Beyond C1C^1, affirmative answers are known in various conservative, low-dimensional and nonuniformly hyperbolic settings.

The obstacle for r≥2r\ge2 is that the perturbation must stay small together with rr derivatives, which drastically limits the room available to move orbits. General proofs have been claimed [Gao2022ClosingClaim], but no such argument is accepted, and the conjecture stands open in every regularity r≥2r\ge2.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.