Entropy of Hamiltonian Flows on Restricted Contact-Type Hypersurfaces

Definitions and mathematical statement

Does positive topological entropy for the Hamiltonian flow on a fiberwise star-shaped hypersurface extend to every hypersurface of restricted contact type?

In a standard exact symplectic setting $(W,\omega={\,\mathrm{d}}\lambda)$, a hypersurface $\Sigma$ is of restricted contact type when a global primitive of $\omega$ restricts to a contact form on $\Sigma$; if a fixed primitive is required, that must be specified. For $W=T^*Q$ with its canonical Liouville structure, fiberwise star-shaped ordinarily means transversality to the radial Liouville vector field in the cotangent fibers, with the relevant radial-side convention. These definitions explain the usual setting but the source does not actually identify $W$, $Q$, or the allowed primitives.

For a regular energy hypersurface $\Sigma=H^{-1}(e)$, write $\varphi_H^t$ for the Hamiltonian flow restricted to $\Sigma$. On a compact invariant hypersurface, the usual entropy quantity is $h_{\mathrm{top}}(\varphi_H^1)$. The requested extension asks whether the positive-entropy conclusion \[ h_{\mathrm{top}}(\varphi_H^1)>0 \] known in the intended fiberwise star-shaped setting continues to hold when that geometric condition is replaced by restricted contact type, while retaining all other hypotheses of the original result.

Those other hypotheses are not supplied. In particular, the ambient manifold, assumptions on the base topology or dynamics, compactness and regularity of the hypersurface, and quantifier over Hamiltonians or time parametrizations are missing. The question cannot be interpreted as asserting positive entropy on every restricted-contact-type hypersurface in every symplectic manifold. No base-group growth, dimension, or geometric assumption is invented to make that statement viable. The distinction between a fixed primitive and existence of some global primitive also requires source review.

The dated assessment describes this entry as only an open-problem marker with no mathematical claim beyond a heading, whereas the current record contains the explicit extension question above. That mismatch is retained as a source-curation issue rather than used to discard the supplied question.

Short English statement

The source asks whether a positive-entropy result for fiberwise star-shaped energy surfaces remains true for restricted-contact-type surfaces. The original ambient and dynamical hypotheses are missing, so it is not a universal entropy assertion for arbitrary symplectic manifolds.

Sources

• Local UnsolvedMath registry, permanent ID 30000534, code catalogue entry, "Entropy of Hamiltonian Flows on Restricted Contact-Type Hypersurfaces." Proposed year 2006; proposer unspecified.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2007), Oberwolfach Reports, pp. 1991--2058, DOI 10.4171/owr/2006/33; https://doi.org/10.4171/owr/2006/33. The displayed report year 2007 differs from the DOI and proposed year 2006.
• Statement status exact; the supplied verification says the statement was confirmed without textual change.
• Literature triage checked 2026-08-21: open. Its assessment calls the extraction a heading-only open-problem marker, despite the explicit entropy-extension question now present. This discrepancy and the source hypotheses require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.