Smale's 10th Problem: The Pugh Closing Lemma

Smale's 10th Problem: The Pugh Closing Lemma

Definitions and mathematical statement

Let $M$ be a compact smooth manifold and $\operatorname{Diff}^r(M)$ the space of $C^r$ diffeomorphisms with its $C^r$ topology. A point $p$ is nonwandering for $f$ if every neighborhood $V$ of $p$ satisfies $f^n(V)\cap V\ne\varnothing$ for some $n\ge1$. The closing assertion at differentiability order $r$ is \[ \forall f\ \forall p\in\Omega(f)\ \forall\mathcal U\ni f \quad\exists g\in\mathcal U\ \exists n\ge1:\ g^n(p)=p. \] Here $\mathcal U$ is a $C^r$ neighborhood and $\Omega(f)$ the nonwandering set. The question asks for the allowed differentiability orders of the source, in particular arbitrarily high finite orders; an infinite-order version uses the usual smooth topology. It concerns perturbations of diffeomorphisms without an added volume-preserving or symplectic restriction. Closing a point merely near $p$ must not replace periodicity of $p$ in this selected wording.

Formulation and concept references: [S1].

Short English statement

Can an arbitrarily small perturbation in the required differentiability topology turn any nonwandering point of a diffeomorphism into a periodic point?

Research attempt

Exact point closing and source scope.

The assertion for unrestricted higher-regularity diffeomorphisms is UNRESOLVED here. The source [S1, Definition 1.1 and Theorem A] explicitly makes the given point periodic and proves that conclusion for partially hyperbolic maps with one-dimensional center. Its introduction records the general $C^1$ theorem and explains why some local perturbation techniques fail at higher regularity. Neither a Hamiltonian-vector-field counterexample nor a theorem restricted to a one-dimensional center settles the unrestricted diffeomorphism question in the root statement.

We give a precise relocation lemma, explicit smooth subcases, and a quantitative local closing construction. The latter makes the derivative cost visible and identifies what recurrence alone fails to estimate. Throughout, a return time $N$ need not be the least period, since the target only asks $g^N(p)=p$.

Moving a nearby periodic point to the prescribed point.

Fix a coordinate ball about $p$, a smaller concentric ball, and a smooth bump function $\chi$ supported in the former and equal to one on the latter. For $q$ close to $p$, the coordinate map \[ h_q(x)=x+\chi(x)(p-q) \tag{318.1} \] sends $q$ to $p$. Extend it by the identity outside the chart. For $q$ sufficiently close, its derivative differs from the identity by less than one in operator norm. The coordinate map is injective because $|h_q(x)-h_q(y)|\geq(1-\|Dh_q-I\|_\infty)|x-y|$. It is a local diffeomorphism and proper, and agrees with the identity outside a compact set, so is a global diffeomorphism. Equivalently, one can use the associated proper degree-one local diffeomorphism.

All derivatives of $h_q-\mathrm{id}$ of each fixed order are bounded by constants times $|p-q|$, since the bump support is fixed. Thus $h_q\to\mathrm{id}$ in every finite $C^r$ topology and in the smooth topology as $q\to p$. The inverses converge likewise by the inverse derivative formulas.

If $q$ is periodic for a map $F$, put $G=h_qFh_q^{-1}$. Then \[ G^N(p)=h_qF^N(q)=h_q(q)=p. \tag{318.2} \] Composition and inversion are continuous in the stated diffeomorphism topologies, so if $F\to f$ and $q\to p$ this construction has $G\to f$. This proves an exact quantifier bridge: arbitrarily close perturbations with periodic points arbitrarily near $p$, with both approximations simultaneous, imply the selected exact-point formulation. A theorem giving one periodic point somewhere in $M$ does not satisfy that premise.

In particular, if $p\in\overline{\operatorname{Per}(f)}$, then $p$ is $C^r$-closable without changing the conjugacy class of $f$. The modification $318.1$ is not claimed volume preserving; the catalogue imposes no such restriction. Importing the same construction unchanged into the conservative problem would require a different relocation lemma.

Two complete classes at every differentiability order.

For translations $R_\alpha(x)=x+\alpha$ of $\mathbb T^d$, choose rational vectors $\alpha_j\to\alpha$. Each $R_{\alpha_j}$ has a common finite period for all its points. The differences between these maps have constant lifts, so their derivatives of every positive order agree. Hence convergence is $C^\infty$, and every point, not just a point near $p$, becomes periodic. If $f=hR_\alpha h^{-1}$ for a fixed smooth diffeomorphism $h$, the same argument with $hR_{\alpha_j}h^{-1}$ proves the conclusion.

For a hyperbolic toral automorphism induced by an integer matrix $A\in GL(d,\mathbb Z)$, rational points are periodic. Indeed for each denominator $q$, $A$ permutes the finite group $(q^{-1}\mathbb Z/\mathbb Z)^d$, so every point in that group has a finite orbit. Rational points are dense; $318.1$--$318.2$ therefore prove exact-point $C^\infty$ closing for every $p$. Hyperbolicity is not even needed for this rational-grid observation, although it gives another connection to the source's setting.

More generally, the standard Anosov closing/shadowing theorem implies density of periodic points in the nonwandering set for an Anosov diffeomorphism. Taking that theorem as an explicitly external hyperbolic-dynamics input, the relocation lemma yields the selected conclusion for such maps at their given regularity. This deduction does not extend the shadowing theorem to arbitrary nonwandering systems with a center direction.

A local construction with a fully quantified cost.

Consider an orbit segment \[ x_0=p,\ x_i=f^i(p)\ (1\leq i\leq N), \qquad |x_N-p|=\delta \] in a coordinate chart near its two endpoints. Suppose a coordinate ball $B(p,\rho)$ misses $x_1,\ldots,x_{N-1}$, and $\delta<\rho/4$. Choose a fixed smooth bump $\chi_0$, equal to one on the ball of radius $1/2$ and supported in the unit ball, and define \[ h(x)=x+(p-x_N)\chi_0((x-p)/\rho). \tag{318.3} \] Then $h(x_N)=p$ and $h$ is the identity outside $B(p,\rho)$. For a multi-index $\alpha$ with $|\alpha|=j$, \[ \|D^\alpha(h-\mathrm{id})\|_\infty \leq C_j\delta\rho^{-j}. \tag{318.4} \] If $C_1\delta/\rho<1$, the injectivity and properness argument used above shows $h$ is a diffeomorphism. Set $g=h\circ f$. Since $h$ fixes every intermediate image, induction gives $g^i(p)=x_i$ for $0\leq i<N$ and $g^N(p)=p$. For fixed $f$ and a fixed finite order $r$, the chain rule on a compact manifold bounds \[ \|g-f\|_{C^r}\leq C(f,r)\max_{0\leq j\leq r} \delta\rho^{-j}. \tag{318.5} \] The coordinate norms are equivalent to any fixed finite atlas norms; the constants incorporate that atlas and the chosen bump.

Consequently, any recurrent point admitting segments with $\rho_j\to0$ and $\delta_j=o(\rho_j^r)$ under these separation conditions is $C^r$-closable by this elementary method. If the ratios vanish at every fixed derivative order along suitable segments, it gives smooth closing as well. A basic smooth neighborhood constrains only finitely many seminorms, so one chooses a segment adequate for the largest required order; one need not control an infinite list by one uniform bound.

Nonwandering is weaker than the recurrence used above.

The definition of nonwandering gives points $q_j\to p$ and times $N_j$ with $f^{N_j}(q_j)\to p$. It does not say that the orbit of $p$ itself returns. If a construction closes these $q_j$ for maps $F_j\to f$, then the relocation lemma closes $p$ exactly. This shows how to use nonwandering returns without silently upgrading them to recurrence of the specified point.

However, even for a recurrent point the endpoint distance $\delta$ alone says nothing about the radius of a ball avoiding all the intermediate orbit points. Other visits can force $\rho$ to be comparable to, or much smaller than, the endpoint error. The data $\delta_j\to0$ and $\rho_j\to0$ do not imply $\delta_j\rho_j^{-r}\to0$.

For a numerical scaling diagnostic, take $\rho_j=1/j$ and $\delta_j=\rho_j^2$. The first derivative cost in $318.4$ tends to zero, while the second derivative cost stays of order one and higher costs increase. These are possible relative scales, not a constructed counterexample diffeomorphism. They refute the inference from a small displacement to a small high-derivative perturbation.

Why smoothing a $C^1$ closing perturbation is insufficient.

Suppose a $C^1$ theorem supplies a nearby map with a periodic point. Approximating that map by smooth maps can preserve a hyperbolic periodic orbit if hyperbolicity is available, but its distance from the original $f$ in $C^r$ is still uncontrolled. Smoothness of an approximant is not smallness of its high derivatives relative to $f$. Formula $318.4$ gives an explicit family of smooth modifications exhibiting precisely this mismatch.

Nor can one perturb just the last return if the support intersects earlier images: then $g^i(p)=f^i(p)$ may fail before time $N$, invalidating the asserted closing identity. A valid multi-step construction must track all altered intermediate points and divide the correction so that every required derivative stays small. That is substantive dynamical work, not a matter of choosing a smaller bump after the fact.

Verification and outcome.

The finite diagnostics enumerate rational torus orbits exactly on finite grids, check the conjugacy identity for finite permutations, and record exact rational costs $\delta\rho^{-r}$ in the displayed scaling example. They do not simulate or claim shadowing for an arbitrary diffeomorphism. The relocation and support arguments above establish the actual exact-point claims.

We have proved the selected formulation for translations, their smooth conjugates, toral automorphisms and, using the declared shadowing input, Anosov maps. We also proved a quantitative sufficient return condition. The missing general step is to exploit arbitrary nonwandering returns while overcoming their support/derivative costs in the prescribed $C^r$ topology. The unrestricted higher-order closing assertion remains UNRESOLVED here.

Sources

• S1 Shaobo Gan and Yi Shi, C^r-Closing lemma for partially hyperbolic diffeomorphisms with 1D-center bundle, Problem 1. https://arxiv.org/html/2004.06855v2. Formulation source.