Pugh–Shub Stable Ergodicity (C^r Density in Partial Hyperbolicity)

Pugh–Shub Stable Ergodicity (C^r Density in Partial Hyperbolicity)

Definitions and mathematical statement

Let $(M,m)$ be a compact connected boundaryless smooth manifold with smooth probability volume. Partial hyperbolicity is a continuous invariant splitting $TM=E^u\oplus E^c\oplus E^s$ with nonzero $E^u,E^s$, possible zero center, and a common iterate $N$ such that unstable unit vectors expand by more than two, stable ones contract below one-half, and earlier nonzero bundles dominate later ones by a factor greater than two at the same base point. A map $g$ is stably ergodic here if a $C^1$ neighborhood of it has every $C^2$ volume-preserving member ergodic for $m$, meaning every invariant measurable set has measure zero or one. For every $r\in[2,\infty]$, the assertion is density of these $g$ in $\operatorname{PH}_m^r(M)$ in the relative $C^r$ topology. Nonintegral finite $r$ uses Hölder regularity. The density topology and the stability topology are intentionally different.

Formulation and concept references: [S1].

Short English statement

Can every volume-preserving partially hyperbolic map be approximated in its full $C^r$ topology by one whose nearby $C^2$ conservative maps remain ergodic?

Research attempt

Two different topologies occur in the question.

The inspected source [S1, Theorem A and Theorem A'] proves $C^1$ density of stable ergodicity among the stated partially hyperbolic conservative maps. This is substantial progress, but the selected assertion asks for approximation in the relative $C^r$ topology, for every $r\geq2$, while retaining $C^1$ stability against $C^2$ conservative perturbations. We do not upgrade the source's density topology by changing the regularity label of the approximating maps.

The partial results below give explicit hyperbolic examples, prove ergodicity but not stable ergodicity in a natural center family, and identify the quantitative perturbation and accessibility issues left by that family. The source's deep perturbation and foliation results are kept distinct from the elementary Fourier and conjugacy calculations supplied here.

An explicit ergodic uniformly hyperbolic map.

On $\mathbb T^2=\mathbb R^2/\mathbb Z^2$, consider \[ A=\begin{pmatrix}2&1\\1&1\end{pmatrix}. \] It has determinant one and eigenvalues $\lambda=(3+\sqrt5)/2>1$ and $\lambda^{-1}<1$. The constant eigendirections provide a stable and unstable splitting, and some common iterate meets the numerical inequalities in the definition. The map preserves normalized area because its determinant is one.

Here is a direct proof of ergodicity. If $f\in L^2(\mathbb T^2)$ is invariant, its Fourier coefficients are constant along every orbit of $A^{\mathsf T}$ on $\mathbb Z^2$. A nonzero integer vector cannot have a finite orbit: that would give $((A^{\mathsf T})^k-I)v=0$ for some $k>0$, whereas neither $\lambda^k$ nor $\lambda^{-k}$ is one. Every nonzero orbit is therefore infinite. Square summability of Fourier coefficients forces the constant coefficient on each such orbit to be zero. Thus every invariant $L^2$ function is constant. Applying this to invariant-set indicators proves ergodicity.

This argument proves the single map's ergodicity without an unstated appeal to topological transitivity. Stable ergodicity requires additional control of nonlinear perturbations.

The Hopf criterion and what its use imports.

For a $C^2$ conservative Anosov diffeomorphism, the stable and unstable foliations are absolutely continuous and have local product structure. We use these standard hyperbolic-dynamics facts as external input, not as consequences of the Fourier proof. They are the kind of foliation input explicitly discussed in [S1]. Given them, the ergodicity argument can be described precisely.

Take a continuous observable $\phi$. Birkhoff's theorem gives forward and backward averages on a full-measure set, and these averages agree almost everywhere, as both equal conditional expectation onto the invariant sigma-algebra. Points on the same stable leaf have the same forward average whenever defined: their forward distances tend to zero, so uniform continuity of $\phi$ makes the difference of the Cesaro averages tend to zero. Points on an unstable leaf have the same backward average by the same reason in reverse time.

Absolute continuity allows these full-measure statements to pass to almost every leaf plaque. In a local product box, a function that is essentially constant on both transverse plaque families is constant almost everywhere: in product coordinates it is a function of the first coordinate alone and also of the second alone, and Fubini forces a common constant. The local product measure is equivalent to the smooth volume measure by the stated foliation input. Overlapping boxes have the same constant, and connectedness of the manifold gives one global constant.

Applying this to a countable dense family of continuous observables shows that their invariant conditional expectations are constants. Density then gives the same for all $L^2$ observables, proving ergodicity. The step invoking absolute continuity is important: topological accessibility of individual points alone would not justify the passage through almost-everywhere Birkhoff sets.

The Anosov property is $C^1$ open. For the displayed linear map, this can be seen from invariant cones around the two eigenlines: in an adapted norm the expansion/contraction factors are $\lambda,\lambda^{-1}$, and sufficiently small derivative perturbations preserve slightly wider cones with strict factors. Iteration of the cone graph maps produces invariant stable and unstable subbundles. Thus every sufficiently $C^1$-close $C^2$ conservative map is again Anosov, and the preceding Hopf argument proves its ergodicity. This verifies stable ergodicity of this example in the exact stability convention of the question.

More generally the same standard Anosov inputs settle the zero-center subcase. The difficulty in the catalogue question lies in permitting a genuine center bundle, where strong stable and unstable plaques no longer form a full-dimensional local product chart.

A center family that is ergodic but never stably ergodic.

On $\mathbb T^2\times\mathbb T$, let \[ F_\alpha(x,\theta)=(Ax,\theta+\alpha). \tag{388.1} \] Its derivative has the splitting into the expanding and contracting eigenlines of $A$ and the vertical center line. The center multiplier is one. A common large iterate has unstable expansion greater than two, stable contraction less than one half, and the required domination over and under the center. Thus it belongs to the selected partially hyperbolic class for every $\alpha$ and is smooth and volume preserving.

If $\alpha$ is irrational, $F_\alpha$ is ergodic. For an invariant $L^2$ function, its coefficients indexed by $(k,\ell)\in \mathbb Z^2\times\mathbb Z$ have constant moduli along the $A^{\mathsf T}$ orbit of $k$, with a unit complex phase from $e^{2\pi i\ell\alpha}$. For $k\ne0$ the orbit is infinite, so square summability again forces vanishing. At $k=0$, invariance requires the coefficient to equal its multiple by $e^{2\pi i\ell\alpha}$, which forces it to vanish for $\ell\ne0$. Only the constant coefficient remains.

If $\alpha=a/q$ is rational, the nonconstant function $e^{2\pi iq\theta}$ is invariant, so the map is not ergodic. Choose rationals $\alpha_j\to\alpha$. Then $F_{\alpha_j}\to F_\alpha$ in every $C^r$ topology, including $C^\infty$, because the difference is only a constant translation in the center coordinate. Therefore no map in the family $388.1$ is stably ergodic, even when it is ergodic.

This is not a counterexample to density in the whole partially hyperbolic space: perturbations outside this product family may be stably ergodic. It proves that proving ergodicity on a dense set of rotation parameters is much weaker than the desired stable-ergodicity assertion.

The obstruction is visible in strong accessibility.

For $388.1$, stable and unstable leaves remain in a fixed circle coordinate. A path made of finitely many strong stable or strong unstable pieces therefore cannot change $\theta$. In the base, the two eigendirections span the tangent plane; concatenating pieces connects points in the same horizontal torus. Thus the accessibility classes are precisely $\mathbb T^2\times\{\theta\}$.

The simple Hopf argument above can at most show that an invariant average is constant along such a class. It has no mechanism to identify averages on different center fibers. For irrational rotations the Fourier argument supplies ergodicity by a separate mechanism, but rational approximation shows that mechanism is not stable. Hence this family provides a concrete test for a proof that claims to deduce stable ergodicity from expansion and contraction alone.

A nonlinear-looking perturbation can still be a product in disguise.

Let $h:\mathbb T^2\to\mathbb R$ be smooth and set $G_h(x,\theta)=(x,\theta+h(x))$. It is a smooth volume-preserving diffeomorphism. Direct calculation gives \[ G_hF_\alpha G_h^{-1}(x,\theta) =(Ax,\theta+\alpha+h(Ax)-h(x)). \tag{388.2} \] Thus a nonconstant fiber cocycle of the form $\phi(x)=\alpha+h(Ax)-h(x)$ does not create genuinely new ergodic behavior. It is conjugate to the product. Its stable and unstable geometry may look tilted in the original coordinates, but the invariant accessibility foliation is just the image of the horizontal tori under $G_h$.

Such a skew product is still not stably ergodic: replacing $\alpha$ by a nearby rational while keeping $h$ fixed gives arbitrarily smooth-close nonergodic conjugate maps. For small $h$ it is an explicit perturbation within the partially hyperbolic neighborhood of the product. More generally partial hyperbolicity is invariant under this smooth conjugacy after taking an appropriate common iterate and equivalent metric.

For any periodic base point $A^q x=x$, a cocycle of this form satisfies the telescoping identity \[ \sum_{j=0}^{q-1}(\phi(A^jx)-\alpha)=0. \tag{388.3} \] Producing a perturbation that violates such an identity for a chosen periodic orbit rules out this particular coboundary description. It does not by itself prove accessibility or stable ergodicity, but it is a useful necessary stress test for a proposed construction. A proof cannot claim to break the product obstruction while using only perturbations of the form $388.2$.

Why C1-small perturbations do not solve the selected approximation.

The topology issue is quantitative. On a torus coordinate strip, choose a smooth bump $\psi$ supported in a short interval and, for a small scale $\varepsilon$, form the volume-preserving shear \[ S_\varepsilon(x,y)=(x+\varepsilon^r\psi(y/\varepsilon),y), \tag{388.4} \] with the bump periodically extended from its small support. Its Jacobian determinant is identically one. For an integer $r\geq2$, its displacement tends to zero in $C^1$ at rate at most $\varepsilon^{r-1}$, while its $r$-th derivative contains $\psi^{(r)}(y/\varepsilon)$ and need not tend to zero at all. At noninteger $r=m+\alpha$, $0<\alpha<1$, the amplitude $\varepsilon^{m+\alpha}$ likewise keeps the $\alpha$-Holder seminorm of the $m$-th derivative at constant scale while the $C^1$ displacement tends to zero.

These are explicit conservative perturbations showing that a $C^1$ error bound does not control the prescribed higher topology. Making a perturbation smooth does not change this fact. The source's theorem supplies smoothness of the approximants in an appropriate class but its closeness conclusion is still only in $C^1$. In the $C^\infty$ topology, each neighborhood controls a finite collection of derivative seminorms; an argument must meet whichever finite collection is prescribed, not just the first one.

One could try to spread a localized perturbation over a larger region to reduce its high derivatives. That may alter the orbit segments and invariant geometry it was meant to control. The construction would need simultaneous estimates for the desired ergodic mechanism and every derivative specified by the neighborhood. No such estimate is supplied by the cone calculation or the product-cocycle algebra above.

Diagnostics and outcome.

The finite script checks the cat-map determinant and trace, numerically checks the telescoping identity on sample periodic orbits, and records the derived derivative-scaling exponents of the conservative shear. The conjugacy is proved directly above. These tests audit the concrete examples; they do not certify a perturbation theorem for arbitrary center dimension.

The proved elementary results distinguish ergodicity, stable ergodicity and the two approximation topologies. With the stated standard Anosov foliation input, the zero-center subcase is settled. For a general partially hyperbolic map, the first missing step is a perturbation in an arbitrary prescribed $C^r$ neighborhood that creates a robust ergodic mechanism across the center directions. The $C^1$ density theorem is not that step. The selected full $C^r$ density assertion remains unresolved by this attempt.

Sources

• S1 Pugh–Shub Stable Ergodicity (C^r Density in Partial Hyperbolicity); exact editorial selection using the primary formulations and hypotheses recorded in the source receipts. https://arxiv.org/pdf/1709.04983. Formulation source.