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Compactness question for complete surfaces satisfying Condition A

Let MM be a complete surface and let f ⁣:M→Mf\colon M\to M be a C1C^{1} diffeomorphism satisfying Condition A: ff has dense periodic points, there is a DfDf-invariant splitting TM=Es⊕EuTM=E^{s}\oplus E^{u} into one-dimensional subbundles with constants C>0C>0 and 0<λ<10<\lambda<1 such that ∥Dfn∣Es∥≤Cλn\lVert Df^{n}|_{E^{s}}\rVert\le C\lambda^{n} and ∥Df−n∣Eu∥≤Cλn\lVert Df^{-n}|_{E^{u}}\rVert\le C\lambda^{n} for every n≥0n\ge 0, and the invariant line fields uniquely integrate to transverse stable and unstable foliations. Must MM be compact?

References

Primary source

arXiv

Additional references

dynamical-systemsgeometrytopology
Posted · edited

Progress summary

Refreshed
Claimed progress

A new paper narrows the possibilities for finite-type examples, but the general compactness question remains open.

The question asks whether a complete surface with the stated Anosov-like structure must be compact, equivalently a two-torus under the relevant hypotheses. It is identified with Question 3.13 of a 2018 source cited by the literature.

Known results

  • A C1C^{1} uniform Anosov diffeomorphism on a complete surface with dense periodic points admits Margulis measures on stable and unstable leaves.
  • Global product structure of the stable and unstable foliations forces the surface to be closed.
  • The corresponding argument excludes open surfaces with fundamental group trivial or Z\mathbb{Z}.

September 2026 finite-type advance

Raúl Ures and Tongyao Yu's paper rules out periodic isolated planar ends and shows that the finite-topological-type case is the two-torus. The report explicitly presents this as a partial answer, not a resolution of the general question.

Current status (as of September 2026): The finite-topological-type case is reported to reduce to the two-torus and periodic isolated planar ends are excluded, but the general compactness question remains open.

Sources

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