$C^r$ Stability Conjecture

Structural stability is a global topological property: a dynamical system is structurally stable if all nearby systems have the same orbit structure, up to continuous reparametrisation. Hyperbolicity is a local differential property: the tangent bundle over the recurrent set splits into uniformly contracting and expanding directions. That these two conditions should be equivalent is one of the deepest principles in smooth dynamics.

The problem is rated L3 on UnsolvedMath and sits at the heart of the global theory of smooth dynamical systems. The case $r = 1$ is resolved. The case $r \geq 2$ is open, and even basic consequences of structural stability that are elementary for $r = 1$ remain unknown for $r = 2$.

Key Definitions #

Structural stability. A diffeomorphism $f \in \mathrm{Diff}^r(M)$ is $C^r$-structurally stable if there exists a $C^r$-neighborhood $\mathcal{U}$ of $f$ such that every $g \in \mathcal{U}$ is topologically conjugate to $f$: there is a homeomorphism $h : M \to M$ with $h \circ f = g \circ h$. The system is therefore robust under $C^r$-small perturbations in the strongest possible sense: topology, not just orbit counts, is preserved.

Axiom A. The diffeomorphism $f$ satisfies Axiom A if:

• the non-wandering set $\Omega(f)$ is hyperbolic: there is a $Df$-invariant splitting $T_x M = E^s_x \oplus E^u_x$ over $\Omega(f)$ with uniform exponential contraction on $E^s$ and expansion on $E^u$;
• the periodic points of $f$ are dense in $\Omega(f)$.

Strong Transversality Condition (STC). For every $x, y \in \Omega(f)$, the stable manifold $W^s(x)$ and the unstable manifold $W^u(y)$ intersect transversally. Tangential intersections, namely homoclinic or heteroclinic tangencies, are forbidden.

Together, Axiom A and the STC constitute what is usually meant by saying $f$ is hyperbolic in the sense of the stability conjecture.

The Two Directions #

The conjecture, as an equivalence, has an easy direction and a hard direction.

Structural stability follows from hyperbolicity (the easy direction). Robbin (1971) proved this for $C^2$ diffeomorphisms; Robinson (1976) extended it to $C^1$. Both proofs use the implicit function theorem on an appropriate space of conjugacies, and work for all $r \geq 1$ since Axiom A + STC is the hypothesis.

Hyperbolicity follows from structural stability (the hard direction) is the conjecture itself. It requires understanding what structural stability forces on the dynamics, ruling out every non-hyperbolic mechanism compatible with stability. This is where the difficulty lies, and where the gap between $r = 1$ and $r \geq 2$ opens.

The $C^1$ Case: Mañé’s Theorem #

The $C^1$ stability conjecture was fully proved by Mañé in 1987.

The proof, published in Publ. Math. IHÉS 66 (1987), 161–210, is a tour de force of $C^1$ perturbation theory. It rests on several tools that are available only in the $C^1$ topology:

• Pugh’s $C^1$ closing lemma (1967): Given a non-wandering point $x$ of $f$, one can make an arbitrarily small $C^1$ perturbation of $f$ to create a periodic orbit passing near $x$. This is the essential mechanism for showing that periodic points are dense in $\Omega(f)$.

• Mañé’s ergodic closing lemma (1982): A more refined version that controls the Lyapunov exponents of the created periodic orbit, allowing the construction of hyperbolic periodic points that shadow the orbit of an ergodic measure.

• Franks’ lemma (1971): Linear maps along periodic orbits can be prescribed independently (up to $C^1$ conjugacy), allowing one to test whether a given splitting is genuinely hyperbolic or can be destroyed by a small $C^1$ perturbation.

The strategy is to assume structural stability and use these tools to show, step by step, that the non-wandering set must be hyperbolic and that tangencies cannot persist. Mañé had proved the surface case ($\dim M = 2$, $r = 1$) earlier, with the full higher-dimensional result completed in the 1987 paper. Aoki (1992) and Hayashi (1992) subsequently settled the closely related Mañé conjecture on the $C^1$ interior of the set of diffeomorphisms with all hyperbolic periodic points.

The Wall at $r \geq 2$ #

The $C^r$ case for $r \geq 2$ is not merely an incremental extension. The tools that power Mañé’s proof are fundamentally $C^1$ phenomena.

The $C^r$ closing lemma is open for $r \geq 2$. Pugh’s closing lemma fails for $r \geq 2$ in general: Gutierrez showed that the local perturbation argument used for $C^1$ does not work in the $C^2$ topology. A $C^r$ closing lemma is available only for specific classes of diffeomorphisms:

• Conservative (volume-preserving) diffeomorphisms on surfaces: Asaoka–Irie ($C^\infty$, 2015), Cristofaro-Gardiner–Prasad–Zhang (2023).
• Partially hyperbolic diffeomorphisms with one-dimensional center bundle (all $r \geq 2$ including $r = \infty$): Gan–Shi (2022) and the follow-up $C^r$-chain closing lemma of Shi–Wang (Ergodic Theory Dynam. Syst. 44, 2024).

In the absence of a general $C^r$ closing lemma, the first step of Mañé’s proof, showing that periodic points are dense in $\Omega(f)$ under $C^r$ structural stability, is not known for $r \geq 2$.

Mañé himself underscored this gap. In the 1987 paper, immediately after the proof of Theorem A, he writes that for $r > 1$ “not even [being] known whether a $C^2$ structurally stable diffeomorphism has at least one periodic point, it seems, to say the least, difficult to prove that they are dense.”

Franks’ lemma also fails for $r \geq 2$. Controlling linear maps along periodic orbits requires $C^1$ perturbations; in higher regularity the ambient perturbation must be smooth and the constraints on higher derivatives can prevent the desired linear behaviour from being achieved.

Research Directions #

1. The $C^r$ Closing Lemma for General Diffeomorphisms #

The most direct path to the $C^r$ stability conjecture passes through a general $C^r$ closing lemma. For $r \geq 2$ this asks: given any non-wandering point of a $C^r$ diffeomorphism, can one make an arbitrarily small $C^r$ perturbation to close the orbit? Answering this in the affirmative for all closed manifolds and all $r \geq 2$ would be a landmark result, and would immediately advance the stability conjecture. The recent progress in conservative surface dynamics (Cristofaro-Gardiner et al., 2023) and partially hyperbolic settings shows the question is not hopeless, but the general dissipative case remains untouched.

2. The Surface Case $\dim M = 2$, $r \geq 2$ #

On surfaces the dynamics is simpler: the non-wandering set has lower-dimensional structure, and the absence of a center bundle means “partially hyperbolic” reduces to “hyperbolic.” Mañé settled the surface case for $r = 1$. The $C^r$ stability conjecture for surfaces and $r \geq 2$ is already an important open target and may be the most accessible subcase. Recent $C^\infty$ closing lemmas for conservative surface diffeomorphisms (Asaoka–Irie) suggest that the conservative surface case may be reachable.

3. Partially Hyperbolic Diffeomorphisms #

A diffeomorphism is partially hyperbolic if the tangent bundle splits as $TM = E^{ss} \oplus E^c \oplus E^{uu}$ with uniform contraction on $E^{ss}$, uniform expansion on $E^{uu}$, and an intermediate “center” bundle $E^c$. For these systems, Gan–Shi (2022) and Shi–Wang (2024) have established $C^r$ closing and chain-closing lemmas when $\dim E^c = 1$. The question is whether $C^r$-structural stability of a partially hyperbolic diffeomorphism forces the center bundle to also become hyperbolic, that is, whether partial hyperbolicity implies full hyperbolicity under stability.

4. The Palis Global Conjecture #

Palis proposed that the complement of the hyperbolic diffeomorphisms is exactly the closure of systems exhibiting homoclinic tangencies or heteroclinic cycles. This is a positive description of non-hyperbolic dynamics, and is a strengthening of the $C^r$ stability conjecture (it would also characterise what structural stability forbids). In $C^1$ topology this programme is largely complete through Bonatti– Crovisier’s connecting lemma (2004) and related results. For $r \geq 2$ it is wide open, and progress on the Palis conjecture in $C^r$ would likely resolve the stability conjecture as a corollary.

5. Flows and the Vector Field Analogue #

The stability conjecture has a natural analogue for $C^r$ vector fields: a $C^r$-structurally stable flow should satisfy Axiom A and the strong transversality condition. For $r = 1$ this is also proved. For $r \geq 2$ it is open. The vector field setting introduces additional complications from singular points (zeros of the vector field), as Labarca–Pacifico showed that on manifolds with boundary stable flows can fail Axiom A, so the correct formulation may need adaptation. Progress on the diffeomorphism case would likely shed light on the flow case as well.

References #

1. Palis, J. & Smale, S. (1970). Structural stability theorems. Proc. Sympos. Pure Math., 14, 223–231.
2. Robbin, J. W. (1971). A structural stability theorem. Annals of Mathematics, 94(2), 447–493.
3. Robinson, C. (1976). Structural stability of $C^1$ diffeomorphisms. Journal of Differential Equations, 22(1), 28–73.
4. Mañé, R. (1987). A proof of the $C^1$ stability conjecture. Publications Mathématiques de l’IHÉS, 66, 161–210.
5. Aoki, N. (1992). The set of Axiom A diffeomorphisms with no cycles. Bol. Soc. Brasil. Mat., 23(1–2), 21–65.
6. Hayashi, S. (1992). Diffeomorphisms in $\mathcal{F}^1(M)$ satisfy Axiom A. Ergodic Theory Dynam. Systems, 12(2), 233–253.
7. Gan, S. & Shi, Y. (2022). $C^r$-closing lemma for partially hyperbolic diffeomorphisms with 1D-center bundle. Journal of Differential Equations, 334, 337–363.
8. Shi, Y. & Wang, X. (2024). $C^r$-chain closing lemma for certain partially hyperbolic diffeomorphisms. Ergodic Theory Dynam. Systems, 44(7), 1923–1944.
9. Bonatti, C. & Crovisier, S. (2004). Récurrence et généricité. Inventiones Mathematicae, 158(1), 33–104.
10. Berger, P. (2017). Lectures on structural stability in dynamics. arXiv:1703.00092.