# P3138: Positive metric entropy for the standard map

- ID: `P3138`
- Reference: `standard-map-positive-metric-entropy`
- Page: https://theoremdb.org/statements/P3138
- Record maturity: Reviewed problem with recorded work

## 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?

### Context

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.

### Problem setup

- **Definition (standard map).** The displayed map of the two-torus, up to the conventional coordinate ordering.
- **Definition (metric entropy).** Kolmogorov-Sinai entropy computed with invariant Lebesgue measure.
- **Remark.** 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.

### What 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.

## 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. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (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.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: Positive entropy can be created by nearby conservative perturbations, and hyperbolic sets of zero area are known for many parameters.

The exact unresolved remainder is: Positive Lebesgue metric entropy for the exact standard map at any K remains unproved.

A complete resolution must meet the following acceptance conditions:
- Exhibit K≠0 and prove h_Leb(T_K)>0.
- Or prove h_Leb(T_K)=0 for every real K.

### Background and intake notes

- 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.
- The release review checked 2 structured sources on 2026-08-01.
- 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.

### Other known results

- **Claim 2** (supported): Positive entropy can be created by nearby conservative perturbations, and hyperbolic sets of zero area are known for many parameters. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Positive entropy can be created by nearby conservative perturbations, and hyperbolic sets of zero area are known for many parameters. Unresolved remainder: Positive Lebesgue metric entropy for the exact standard map at any K remains unproved. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): Positive Lebesgue metric entropy for the exact standard map at any K remains unproved.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `standard-map-positive-metric-entropy`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://people.math.harvard.edu/~knill/books/KnillDissertation.pdf
   - Also cited at O. Knill, dissertation discussion of the standard-map entropy problem. standard map open problem discussion
   - book; secondary source; checked 2026-08-01
   - Source use: original_summary
   - Records that existence of a parameter with positive metric entropy is open.
   - 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.
2. <a id="reference-2"></a>On positive metric entropy problem, Reconnect project overview. cemeai.icmc.usp.br checked 2026-08-01. Sinai positive-entropy conjecture discussion https://www.cemeai.icmc.usp.br/Reconnect/on-positive-metric-entropy-problem/
   - website; reference source; checked 2026-08-01
   - Source 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.