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$C^r$ Stability Conjecture

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Statement

Definitions and mathematical statement

Let $M$ be a smooth compact manifold without boundary, equipped with a Riemannian norm, and let $r\in\{1,2,\ldots\}\cup\{\infty\}$. A $C^r$ diffeomorphism $f:M\to M$ is $C^r$ structurally stable if some neighbourhood $\mathcal U$ of $f$ in the $C^r$ topology has this property: for every $g\in\mathcal U$ there exists a homeomorphism $h:M\to M$ satisfying $h\circ f=g\circ h$. The conjugating map is not required to be differentiable.

The nonwandering set $\Omega(f)$ consists of points $x$ such that every neighbourhood $U$ of $x$ satisfies $f^j(U)\cap U\ne\varnothing$ for some integer $j\ge1$. It is hyperbolic if the tangent bundle over it has a continuous $Df$-invariant splitting $E^s\oplus E^u$ and constants $C>0$, $0<\lambda<1$ such that, for $j\ge0$, \[ \|Df^j v\|\le C\lambda^j\|v\|\ (v\in E^s),\qquad \|Df^{-j}w\|\le C\lambda^j\|w\|\ (w\in E^u). \] The Axiom A condition additionally requires the periodic points, those with $f^j(x)=x$ for some $j\ge1$, to be dense in $\Omega(f)$.

In its standard stability-conjecture formulation, the question is whether every $C^r$ structurally stable diffeomorphism satisfies Axiom A, particularly for $r>1$. This specifies the catalogue's abbreviated word "hyperbolic"; it does not demand a hyperbolic splitting over all of $M$ (the Anosov condition). Both the topology of perturbations and the regularity of $f$ are fixed by $r$.

Short English statement

Does persistence up to topological conjugacy under small $C^r$ perturbations force hyperbolic nonwandering dynamics with dense periodic points?

Sources

Catalogue entry

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Math-1010-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Math-1010

$C^r$ Stability Conjecture

Definitions and mathematical statement

Let $M$ be a smooth compact manifold without boundary, equipped with a Riemannian norm, and let $r\in\{1,2,\ldots\}\cup\{\infty\}$. A $C^r$ diffeomorphism $f:M\to M$ is $C^r$ structurally stable if some neighbourhood $\mathcal U$ of $f$ in the $C^r$ topology has this property: for every $g\in\mathcal U$ there exists a homeomorphism $h:M\to M$ satisfying $h\circ f=g\circ h$. The conjugating map is not required to be differentiable.

The nonwandering set $\Omega(f)$ consists of points $x$ such that every neighbourhood $U$ of $x$ satisfies $f^j(U)\cap U\ne\varnothing$ for some integer $j\ge1$. It is hyperbolic if the tangent bundle over it has a continuous $Df$-invariant splitting $E^s\oplus E^u$ and constants $C>0$, $0<\lambda<1$ such that, for $j\ge0$, \[ \|Df^j v\|\le C\lambda^j\|v\|\ (v\in E^s),\qquad \|Df^{-j}w\|\le C\lambda^j\|w\|\ (w\in E^u). \] The Axiom A condition additionally requires the periodic points, those with $f^j(x)=x$ for some $j\ge1$, to be dense in $\Omega(f)$.

In its standard stability-conjecture formulation, the question is whether every $C^r$ structurally stable diffeomorphism satisfies Axiom A, particularly for $r>1$. This specifies the catalogue's abbreviated word "hyperbolic"; it does not demand a hyperbolic splitting over all of $M$ (the Anosov condition). Both the topology of perturbations and the regularity of $f$ are fixed by $r$.

Short English statement

Does persistence up to topological conjugacy under small $C^r$ perturbations force hyperbolic nonwandering dynamics with dense periodic points?

Research attempt

Scope and process.
This is a GPT-6 Astra Ultra research attempt dated 1 October 2026, using primary-source browsing and elementary mathematical checks. The canonical definition above is retained. Results below are restricted results or reductions, not a claim of a new solution to the entire catalogue question. No formal proof assistant or external mathematical referee checked this text.
Literature and logical scope.
Mañé's theorem [S3, R1] establishes that $C^1$ structural stability implies Axiom A on closed manifolds. The $r>1$ formulation in [S2] has a stronger topology of allowed perturbations; being stable against that smaller collection does not imply $C^1$ stability. Our work below constructs smooth instabilities and quantifies this topology obstruction. It is not a reproof of Mañé's theorem.
An explicit nonhyperbolic circle example.
Work on $M=\mathbb R/\mathbb Z$. Fix $0<a<1/(2\pi)$ and take the lift \[ F(x)=x+a(1-\cos 2\pi x). \] It satisfies $F(x+1)=F(x)+1$ and $F'(x)=1+2\pi a\sin2\pi x>0$, hence induces a smooth diffeomorphism $f$. Its displacement lies in $[0,2a]\subset[0,1)$ and vanishes exactly at integers, so $f$ has exactly one fixed point. Its multiplier there is one, so that periodic point is not hyperbolic.

For $0<\delta<1-2a$ set $F_\delta(x)=F(x)+\delta$. The derivative is unchanged and its displacement lies strictly between zero and one; therefore the induced $f_\delta$ has no fixed point. These maps converge to $f$ in every $C^r$ topology, including $C^\infty$. Topological conjugacy maps the fixed-point set bijectively onto the fixed-point set, because $h(f(x))=g(h(x))$. Thus $f$ and $f_\delta$ cannot be conjugate. This explicitly proves instability in all regularities for this parabolic example, with no appeal to a perturbation lemma.

A neutral recurrent example without fixed points.
For a rigid circle rotation $R_\alpha(x)=x+\alpha$, the multiplier is one everywhere. If $\alpha$ is rational, every point is periodic; if irrational, no point is periodic, since $q\alpha\notin\mathbb Z$ for every positive $q$. Rational and irrational parameters approximate one another arbitrarily closely, and changing the parameter gives convergence in every $C^r$ norm on lifts. Existence of a periodic point is a conjugacy invariant, so every rigid rotation is structurally unstable in every specified regularity. This checks a different mechanism from the isolated parabolic fixed point above.
Why a $C^1$ perturbation proof does not automatically lift.
In a local coordinate interval let $\phi$ be a nonzero smooth bump supported in $(-1,1)$, and consider a displacement $b_\varepsilon(x)=\varepsilon^2\phi(x/\varepsilon)$. Its $j$th derivative has supremum $\varepsilon^{2-j}\|\phi^{(j)}\|_\infty$. Thus $b_\varepsilon\to0$ in $C^1$, but its second derivative stays of fixed nonzero size, and higher derivatives may diverge. Extending by zero gives smooth local perturbations; for small $\varepsilon$ the derivative change is small enough that a diffeomorphism remains a diffeomorphism. These are legitimate small $C^1$ perturbations which are not small $C^2$ perturbations. A proof that relies on their availability has not met the $r=2$ hypothesis.
Verification and remaining gap.
The fixed-point count used above is on the quotient circle, so we checked that the displacement cannot equal a nonzero integer. Merely checking that it is nonzero as a real number would not have sufficed. The rotation argument uses periodic points of every period, not a change of a chosen numerical rotation parameter alone, which conjugacy could otherwise obscure. The bump calculation makes no claim that every nonhyperbolic orbit can be handled by this particular bump. In higher dimensions the missing step is a perturbation theorem, small in the fixed $C^r$ topology, which turns general failure of Axiom A into a topological conjugacy obstruction. Neither circle construction supplies that theorem or proves density of periodic points in the general nonwandering set.
Final status.
UNRESOLVED for the general higher-regularity target. The $C^1$ case is attributed to existing literature; two explicit smooth families and the precise norm obstruction are independently verified here.

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