An SRB measure for the classical Hénon parameters
Let $H$ be the polynomial diffeomorphism of $\mathbb{R}^2$ $$H(x, y) = (1 - 1.4 x^2 + y,\ 0.3 x).$$ Hénon studied it numerically and observed orbits approaching a strange attractor (1976). An SRB measure is an ergodic invariant probability measure with a positive Lyapunov exponent and absolutely continuous conditional measures on local unstable manifolds. For the family $(x, y) \mapsto (1 - a x^2 + y, b x)$, Benedicks and Carleson proved that for small $b > 0$ there is a strange attractor for a set of $a$ of positive Lebesgue measure (1991), and Benedicks and Young proved that for a set of parameters $(a, b)$ of positive Lebesgue measure the map has a unique SRB measure (1993). Galias and Tucker found periodic sinks of low period at parameters extremely close to $(1.4, 0.3)$ (2015).
Problem. Does $H$ admit a compactly supported ergodic invariant probability measure $\mu$ with a positive largest Lyapunov exponent and absolutely continuous conditional measures on its local unstable manifolds?
Source: Hénon (1976), from numerical experiments at these parameters; the conjecture that the map has an SRB measure is credited to him by Xénelkis de Hénon (2024), §1.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- DS — Dynamical Systems
- Keywords
- AIM Open Applied Problems, Hénon maps, SRB measures, strange attractors
References
@article{henon1976,
author = {H{\'e}non, Michel},
title = {A two-dimensional mapping with a strange attractor},
journal = {Comm. Math. Phys.},
volume = {50},
number = {1},
year = {1976},
pages = {69--77},
doi = {10.1007/BF01608556},
}
@article{dehenon2024,
author = {de H{\'e}non, Julia X{\'e}nelkis},
title = {H{\'e}non maps: a list of open problems},
journal = {Arnold Math. J.},
volume = {10},
number = {4},
year = {2024},
pages = {585--620},
doi = {10.1007/s40598-024-00252-x},
}
@article{benedicks1991,
author = {Benedicks, Michael and Carleson, Lennart},
title = {The dynamics of the {H}{\'e}non map},
journal = {Ann. of Math. (2)},
volume = {133},
number = {1},
year = {1991},
pages = {73--169},
doi = {10.2307/2944326},
}
@article{benedicks1993,
author = {Benedicks, Michael and Young, Lai-Sang},
title = {Sinai--{B}owen--{R}uelle measures for certain {H}{\'e}non maps},
journal = {Invent. Math.},
volume = {112},
year = {1993},
pages = {541--576},
doi = {10.1007/BF01232446},
}
@article{galias2015,
author = {Galias, Zbigniew and Tucker, Warwick},
title = {Is the {H}{\'e}non attractor chaotic?},
journal = {Chaos},
volume = {25},
number = {3},
year = {2015},
pages = {033102},
doi = {10.1063/1.4913945},
}
@misc{aim-classical-henon-srb-measure,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{An SRB measure for the classical Hénon parameters}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/128-classical-henon-srb-measure.md},
note = {Entry 128 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00508, "An SRB measure for the classical Hénon parameters", clopen.solutions.