Math Conjectures

Pugh–Shub stable-ergodicity conjecture

OPENMajorConjectureProposed 1997 · Standard version

Canonical statement

Let MM be a closed connected manifold with a smooth probability volume μ\mu, and fix a Riemannian norm on TMTM. For a linear map AA, write m(A)=inf⁡∥v∥=1∥Av∥\mathfrak m(A)=\inf_{\|v\|=1}\|Av\|. A C2C^2, μ\mu-preserving diffeomorphism ff is partially hyperbolic here if it has a continuous DfDf-invariant splitting into nonzero bundles TM=Es⊕Ec⊕EuTM=E^s\oplus E^c\oplus E^u and an integer N≥1N\ge1 such that, for every x∈Mx\in M,
∥DfxN∣Exs∥<m(DfxN∣Exc)≤∥DfxN∣Exc∥<m(DfxN∣Exu),∥DfxN∣Exs∥<1<m(DfxN∣Exu). \|Df_x^N|_{E_x^s}\|<\mathfrak m(Df_x^N|_{E_x^c})\le\|Df_x^N|_{E_x^c}\|<\mathfrak m(Df_x^N|_{E_x^u}),\qquad \|Df_x^N|_{E_x^s}\|<1<\mathfrak m(Df_x^N|_{E_x^u}).
Among these diffeomorphisms, the stably ergodic ones are C2C^2-dense; stable ergodicity means that every sufficiently C2C^2-near C2C^2, μ\mu-preserving diffeomorphism is ergodic.
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Let \(M\) be a closed connected manifold with a smooth probability volume \(\mu\), and fix a Riemannian norm on \(TM\). For a linear map \(A\), write \(\mathfrak m(A)=\inf_{\|v\|=1}\|Av\|\). A \(C^2\), \(\mu\)-preserving diffeomorphism \(f\) is partially hyperbolic here if it has a continuous \(Df\)-invariant splitting into nonzero bundles \(TM=E^s\oplus E^c\oplus E^u\) and an integer \(N\ge1\) such that, for every \(x\in M\), \[ \|Df_x^N|_{E_x^s}\|<\mathfrak m(Df_x^N|_{E_x^c})\le\|Df_x^N|_{E_x^c}\|<\mathfrak m(Df_x^N|_{E_x^u}),\qquad \|Df_x^N|_{E_x^s}\|<1<\mathfrak m(Df_x^N|_{E_x^u}). \] Among these diffeomorphisms, the stably ergodic ones are \(C^2\)-dense; stable ergodicity means that every sufficiently \(C^2\)-near \(C^2\), \(\mu\)-preserving diffeomorphism is ergodic.
DYN-012:[PughShub1997StableErgodicity][BurnsWilkinson2010StableErgodicity][AvilaCrovisierWilkinson2017PositiveEntropy][LeguilZhang2026PrevalenceStableErgodicity]
Dynamical systemsProbability theory

Notes

Pugh and Shub conjectured in the 1990s that, among C2C^2 volume-preserving partially hyperbolic diffeomorphisms of a closed manifold, the stably ergodic ones — those all of whose sufficiently C2C^2-close volume-preserving perturbations are ergodic — form a C2C^2-dense set [PughShub1997StableErgodicity]. Partial hyperbolicity means the tangent bundle splits invariantly as Es⊕Ec⊕EuE^s\oplus E^c\oplus E^u, with uniform contraction on EsE^s, uniform expansion on EuE^u, and intermediate behaviour on the center; the slogan is that a little hyperbolicity generically yields robust ergodicity.

The program divides into proving that accessibility (joining any two points by paths along stable and unstable leaves) is dense, and that accessibility implies ergodicity. On the latter, Burns and Wilkinson proved that accessibility together with center bunching yields ergodicity for C2C^2 volume-preserving partially hyperbolic systems [BurnsWilkinson2010StableErgodicity]. Density of stable ergodicity is known in several center dimensions and isotopy classes; a C1C^1 analogue of the density statement was obtained by Avila, Crovisier and Wilkinson [AvilaCrovisierWilkinson2017PositiveEntropy], and prevalence-type results hold for certain classes in higher regularity [LeguilZhang2026PrevalenceStableErgodicity].

What blocks a global C2C^2-density theorem is general center dynamics, for which no ergodicity criterion of comparable power exists; the conjecture remains open.

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.