[#P3138] Positive metric entropy for the standard map
Contents
Problem. Does there exist a nonzero real parameter \(K\) for which the Chirikov standard map \(T_K(x,y)=(x+y+K\sin x,\,y+K\sin x)\pmod{2\pi}\) has positive Kolmogorov-Sinai entropy with respect to Lebesgue area?
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Definitions and notation
1Context
Known frontier: Positive entropy can be created by nearby conservative perturbations, and hyperbolic sets of zero area are known for many parameters. Open boundary: Positive Lebesgue metric entropy for the exact standard map at any K remains unproved.
2Problem setup
Definition 1 (standard map). The displayed map of the two-torus, up to the conventional coordinate ordering.
Definition 2 (metric entropy). Kolmogorov-Sinai entropy computed with invariant Lebesgue measure.
Remark 1. The standard map is an explicit analytic area-preserving twist map. Numerics show large chaotic seas, yet proving a positive-area set with nonzero Lyapunov exponent for any parameter has resisted current methods.
3What counts as a solution
- Exhibit K≠0 and prove h_Leb(T_K)>0.
- Or prove h_Leb(T_K)=0 for every real K.
1Status
Saved packet · August 1, 2026
Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: Positive entropy can be created by nearby conservative perturbations, and hyperbolic sets of zero area are known for many parameters. Exact unresolved remainder: Positive Lebesgue metric entropy for the exact standard map at any K remains unproved.[1][2]
1Packet records
Recent contributions
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Notes and companion material
Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: Positive entropy can be created by nearby conservative perturbations, and hyperbolic sets of zero area are known for many parameters. Exact unresolved remainder: Positive Lebesgue metric entropy for the exact standard map at any K remains unproved.
- Equivalent-formulation queries: standard map positive metric entropy exists parameter open 2026; Chirikov standard map Lebesgue entropy positive proof
- Strongest checked neighboring result: Positive entropy can be created by nearby conservative perturbations, and hyperbolic sets of zero area are known for many parameters.
- Exact unresolved remainder: Positive Lebesgue metric entropy for the exact standard map at any K remains unproved.
How the 4 records connect
ProblemPositive metric entropy for the standard map
All 3 recorded relations between these records and the problem
2See also
- Hilbert's sixteenth problem, second partdynamical systems
- Birkhoff conjecture for integrable convex billiardsdynamical systems
- Persistent exponential stretching of a material line in two-dimensional Euler flowdynamical systems
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Cite this problem statement
Cite the original sources separately.
“Positive metric entropy for the standard map.” TheoremDB. P3138. Problem statement; statement text SHA-256 0ce6b1bfdd7a586b3105e6966ceacd14f631092309ed6cf91909e8d0bd3b7e1a. https://theoremdb.org/statement/?ref=P3138
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title = {{Positive metric entropy for the standard map}},
howpublished = {TheoremDB},
note = {Problem statement; statement text SHA-256 0ce6b1bfdd7a586b3105e6966ceacd14f631092309ed6cf91909e8d0bd3b7e1a},
url = {https://theoremdb.org/statement/?ref=P3138}
}Plain text: Built Markdown snapshot
This problem includes 4 records joined by 3 typed links, sourced from people.math.harvard.edu[1], current as of August 1, 2026.
1References
- Packet source. O. Knill, dissertation discussion of the standard-map entropy problem. standard map open problem discussion. people.math.harvard.edu checked 2026-08-01. standard map open problem discussion. ↗book · secondary source · checked 2026-08-01Source use: original summary.Records that existence of a parameter with positive metric entropy is open.Also cited at O. Knill, dissertation discussion of the standard-map entropy problem. standard map open problem discussion.Source used to assess the problem's recorded status.For Positive metric entropy for the standard map: This is the dated publication status for the canonical target Positive metric entropy for the standard map.Source named by the research packet.
- On positive metric entropy problem, Reconnect project overview. cemeai.icmc.usp.br checked 2026-08-01. Sinai positive-entropy conjecture discussion. ↗website · reference source · checked 2026-08-01Source use: original summary.Describes current weak approximation results and the unresolved exact standard-map problem.Source used to assess the problem's recorded status.For Positive metric entropy for the standard map: Describes current weak approximation results and the unresolved exact standard-map problem.
Original TheoremDB editorial statement and source synthesis; external works are used for citation only.
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