The good, the bad, and the ugly

Definitions and mathematical statement

Let $\Phi^t$ be a Hamiltonian flow on the intended invariant phase space $\Sigma$, equipped with the invariant Liouville measure $\mu$ described by the source. The dimension, smoothness, compactness, finiteness of $\mu$, and whether $\Sigma$ is the full symplectic manifold or an energy surface are not specified.

Write $G$ for the source's "good set": the maximal invariant subset on which the invariant Liouville measure is almost periodic. The record does not define almost periodicity of a measure or the sense of maximality. One usual measure-theoretic possibility would be precompactness of the Koopman orbit of every $f\in L^2$ for the restricted flow, but that meaning is not imposed here.

Write $B$ for the "bad set," described as the Pesin set with a positive Lyapunov exponent. At a Lyapunov-regular point where exponents are defined, its stated positivity criterion is \[ x\in B\quad\Longleftrightarrow\quad \text{at least one Lyapunov exponent of }D\Phi^t_x \text{ is strictly positive}. \] The assumptions ensuring Lyapunov regularity almost everywhere and the intended Pesin-set convention are not supplied. In particular the source does not say whether it requires more than the existence of one positive exponent.

The "ugly set" is \[ U=\Sigma\setminus(G\cup B). \] The question is whether there exists a Hamiltonian system in the intended class for which $U$ is measurable and \[ \mu(U)>0. \] This retains the question relative to the source's good and bad sets. A fully determined existence problem needs the missing almost-periodicity, maximality, regularity, and Liouville-measure conventions fixed; no alternative definition of $G$ or $B$ is silently substituted.

Short English statement

Can a positive-Liouville-measure part of a Hamiltonian system lie outside both its maximal almost-periodic region and its region with a positive Lyapunov exponent? The supplied research summary reports no verified direct answer and calls the question open. It says related work on robust non-hyperbolic measures does not answer this Hamiltonian question for the specified Liouville measure.

The source is a 2000 problem list, and provenance records no later authoritative resolution check. The OPEN-TRIAGE label therefore remains a supplied assessment, not a new claim of verified current status.

Sources

• Local UnsolvedMath registry, permanent ID 4200010, source code catalogue entry, "The good, the bad, and the ugly," proposed by Oliver Knill, 2000. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Oliver Knill, Open problems in Hamiltonian dynamics (2000), item 10, "THE GOOD THE BAD AND THE UGLY," https://people.math.harvard.edu/~knill/seminars/intr/; HTML extraction, accessed 2026-07-29.
• The supplied research summary mentions related robust non-hyperbolic measures but gives no primary reference or precise theorem that answers this Liouville-measure formulation.
• Source provenance marks status evidence NEEDS_REVIEW and rights NEEDS_REVIEW: the source presents the item as open, no later authoritative resolution check is recorded, and redistribution permission was not established despite verified public access. This qualification is separate from the dataset licence.