[#P3110] Generic analytic Arnold diffusion in a priori stable systems
Problem. For a real-analytic steep integrable Hamiltonian \(h(I)\) with at least three degrees of freedom, is Arnold diffusion present for a generic sufficiently small real-analytic perturbation \(H_\varepsilon(I,\theta)=h(I)+\varepsilon f(I,\theta)\), in the sense that some orbit changes its action by an amount independent of \(\varepsilon\)?
1Context
Known frontier: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Open boundary: A general analytic genericity theorem for a priori stable steep systems remains open.
2Problem setup
Definition 1 (a priori stable). The unperturbed system is integrable without a built-in hyperbolic subsystem.
Definition 2 (Arnold diffusion). An orbit has order-one drift in action variables under an arbitrarily small perturbation.
Remark 1. KAM tori occupy large measure and Nekhoroshev theory gives very long stability. In three or more degrees of freedom those tori do not separate the energy surface, leaving thin resonance channels where diffusion is expected.
3What counts as a solution
- State a standard Baire or prevalence notion of genericity and prove order-one diffusion for the corresponding analytic perturbations.
- Or disprove the generic claim by an open or residual obstruction class under the same topology.
1Status
Current status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.[1][2]
1Records
Notes and companion material
Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.
- Equivalent-formulation queries: generic analytic Arnold diffusion a priori stable open problem; Arnold diffusion analytic genericity stable Hamiltonian
- Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes.
- Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.
How the 4 records connect
ProblemGeneric analytic Arnold diffusion in a priori stable systems
2See also
- Positive metric entropy for the standard mapdynamical systems
- Persistent exponential stretching of a material line in two-dimensional Euler flowdynamical systems
- Birkhoff conjecture for integrable convex billiardsdynamical systems
How to cite
TheoremDB contributors, “Generic analytic Arnold diffusion in a priori stable systems,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/generic-analytic-arnold-diffusion-apriori-stableThis page as plain text: generic-analytic-arnold-diffusion-apriori-stable.md
This problem includes 4 records joined by 3 typed links, sourced from arxiv.org[1], current as of August 1, 2026.
1References
- Packet source. Patrick Bernard, “Arnold's Diffusion: from the a priori unstable to the a priori stable case”. arXiv:1203.2893 (2012). a priori stable discussion. ↗preprint · secondary source · arXiv:1203.2893, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Explains the transition to the hard stable case and outlines prospective mechanisms.Also cited at P. Bernard, Arnold's diffusion: from the a priori unstable to the a priori stable case. a priori stable discussion.Source used to assess the problem's recorded status.For Generic analytic Arnold diffusion in a priori stable systems: This is the dated publication status for the canonical target Generic analytic Arnold diffusion in a priori stable systems.Source named by the research packet.
- Qinbo Chen and Rafael de la Llave, “Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems”. Nonlinearity, vol.35, no.4, 2022. DOI 10.1088/1361-6544/ac50bb. arXiv:2103.03847 (2021). main genericity theorem. ↗preprint · primary source · arXiv:2103.03847, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Proves analytic genericity in a broad a priori unstable class, isolating the stable extension.Source used to assess the problem's recorded status.For Generic analytic Arnold diffusion in a priori stable systems: Proves analytic genericity in a broad a priori unstable class, isolating the stable extension.
Original TheoremDB editorial statement and source synthesis; external works are used for citation only.