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Positive Metric Entropy for the Standard Map at Large Parameters
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Abstract
We prove that the standard sine map of the two-dimensional torus has positive metric entropy with respect to normalized area for every sufficiently large positive parameter. This gives a full parameter tail, and hence answers Sinai's positive-parameter-measure conjecture affirmatively.
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Notes
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work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Oliver Knill, Positive entropy for Standard Map by Open AI (Quantum Calculus blog, 7 October 2026, updated 10 October 2026). Knill, who worked on this problem for over ten years, reports that the release covers positive entropy for large parameters, notes the problem remains open for smaller parameters, and calls the solution "just plain ugly" based on what he had seen so far.
Formalization
- OpenAI math release, Lean scope note for family 146: Positive metric entropy for the standard map (lean/docs/146.md, release of 6 October 2026). The release's scope note describes Lean statements for positive metric entropy at every sufficiently large parameter, positive Lyapunov exponents on positive area with a derivative-growth limit, and a positive-area ergodic hyperbolic component with a finite cyclic Bernoulli decomposition (Corollary 1.2), linking three comparator files. [AI-generated per source] [family-level]
Possibly relevant literature (found by the agent, not checked)
- O. Knill, The Problem of Positive Kolmogorov-Sinai entropy for the Standard map (arXiv:math/9908014, v1 3 August 1999, v2 25 November 2005). An earlier attempt at the statement of Theorem 1.1 based on Herman's subharmonic-estimate idea; in v2 the author withdraws the incorrect parts and states that the entropy conjecture is open.
- J. De Simoi, On cyclicity one elliptic islands of the Standard family (arXiv:1107.3518, 2011, revised 2012). A large-parameter study of elliptic islands, relevant to the paper's discussion of elliptic islands (Duarte) coexisting with the positive-area hyperbolic component of Corollary 1.2.
- G. Del Magno, J. Lopes Dias, J. P. Gaivão, Lyapunov exponents and rigidity in random billiards and standard maps (arXiv:2601.08814, January 2026, v3 July 2026). Proves nonzero exponents for random additive perturbations of generalized standard maps, related to the paper's comparison with randomly perturbed standard maps (Blumenthal–Xue–Young).
- Ziyu Liu and Yankai Shi, Quantitative exponential mixing for the randomized Chirikov standard map (arXiv:2605.20702, 2026-05-20). Its large-kick exponential mixing theorem for a randomized standard map offers a comparison for the paper's weighted fast-bridge estimate, which concerns the exact deterministic map.
Digests and expositions
- heyprof, Standard map: chaos on genuine area (VisuApp, undated). An interactive exposition of family 146 illustrating the positive-area Lyapunov-exponent claim with phase portraits and numerical area estimates, which the page distinguishes from proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 146
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026,
author = {{OpenAI}},
title = {{Positive Metric Entropy for the Standard Map at Large Parameters}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf}{OAI:Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/146.md. The Lean statements have not been checked by people against the paper's claims.
Family 146: Positive metric entropy for the standard map
Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail.
This is the only paper in this family.