Positive metric entropy for a standard map
On $\mathbb{T}^2 = (\mathbb{R}/2\pi\mathbb{Z})^2$ with normalized Lebesgue measure $m$, the standard map $$T_K(x, y) = (x + y + K \sin x,\ y + K \sin x) \pmod{2\pi}, \qquad K \in \mathbb{R},$$ preserves $m$. Let $h_m(T_K)$ be its Kolmogorov–Sinai entropy with respect to $m$. Gorodetski proved that for topologically generic sufficiently large parameters the stochastic sea of the standard map has full Hausdorff dimension. Berger and Turaev proved that for all sufficiently large or sufficiently small parameters the standard map can be $C^\infty$-approximated by area-preserving diffeomorphisms with positive metric entropy, and Blumenthal, Xue and Young obtained lower bounds on Lyapunov exponents for small random perturbations of it.
Problem. Is there a real $K \ne 0$ with $h_m(T_K) > 0$?
Source: Sinai (1994), p. 144, as a conjecture for a set of parameters of positive Lebesgue measure, in the formulation of Berger and Turaev (2019), Conjecture 0.4; stated for a single parameter by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Lyapunov exponents, metric entropy, standard map
References
@book{sinai1994,
author = {Sinai, Ya. G.},
title = {Topics in Ergodic Theory},
series = {Princeton Mathematical Series},
volume = {44},
publisher = {Princeton University Press},
address = {Princeton, NJ},
year = {1994},
doi = {10.1515/9781400887255},
}
@article{berger2019,
author = {Berger, Pierre and Turaev, Dmitry},
title = {On {H}erman's positive entropy conjecture},
journal = {Adv. Math.},
volume = {349},
year = {2019},
pages = {1234--1288},
doi = {10.1016/j.aim.2019.04.002},
eprint = {1704.02473},
archivePrefix = {arXiv},
}
@article{gorodetski2012,
author = {Gorodetski, Anton},
title = {On stochastic sea of the standard map},
journal = {Comm. Math. Phys.},
volume = {309},
number = {1},
year = {2012},
pages = {155--192},
doi = {10.1007/s00220-011-1365-z},
eprint = {1007.1474},
archivePrefix = {arXiv},
}
@article{blumenthal2017,
author = {Blumenthal, Alex and Xue, Jinxin and Young, Lai-Sang},
title = {Lyapunov exponents for random perturbations of some area-preserving maps including the standard map},
journal = {Ann. of Math. (2)},
volume = {185},
number = {1},
year = {2017},
pages = {285--310},
doi = {10.4007/annals.2017.185.1.5},
eprint = {1701.07583},
archivePrefix = {arXiv},
}
@article{blumenthal2021,
author = {Blumenthal, Alex},
title = {Statistical properties for compositions of standard maps with increasing coefficient},
journal = {Ergodic Theory Dynam. Systems},
volume = {41},
number = {4},
year = {2021},
pages = {981--1024},
doi = {10.1017/etds.2019.115},
eprint = {1710.09058},
archivePrefix = {arXiv},
}
@misc{aim-standard-map-metric-entropy,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Positive metric entropy for a standard map}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/065-standard-map-metric-entropy.md},
note = {Entry 065 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
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.
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/paper.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00720, "Positive metric entropy for a standard map", clopen.solutions.