Positive-measure coexistence of attractors in the classical Hénon family beyond the small-dissipation regime, with quantitative parameter-measure bounds

Benedicks and Palmisano (Bull. Braz. Math. Soc., 2023) proved, for the classical Hénon family $f_{a,b}:(x,y)\mapsto(1-ax^2+y,\,bx)$, that positive two-dimensional Lebesgue measure sets of parameters carry at least $k$ attractive periodic orbits together with one Benedicks–Carleson-type strange attractor and — their main result — two coexisting strange attractors, and they established the existence of parameters with infinitely many sinks (the Newhouse phenomenon) as an existence result, without a positive-measure claim. All of these positive-measure statements are constructed only in a small-dissipation regime $b<b_0$, with $b_0$ an unspecified sufficiently small constant, and the paper provides no quantitative lower bounds on the measures of the good parameter sets; moreover, the uniformity-in-$b$ (b-derivative distortion) estimates that the authors themselves flagged as needed for the two-strange-attractor construction are absent from both the preprint and the published version. How far beyond the small-$b$ regime do these coexistence results extend in the classical Hénon parameter domain $0<a<2$, $0<b<1$; what explicit lower bounds on the measures of the parameter sets can be given; and can the positive two-dimensional measure claim of the two-strange-attractor theorem be completely justified.

Background

The Hénon family is the two-parameter family of plane maps $f_{a,b}(x,y)=(1-ax^2+y,\,bx)$ with $0<a<2$, $0<b<1$, introduced by M. Hénon (1976) as a minimal model of a dissipative (area-contracting, since the Jacobian determinant equals $-b$) system with complicated observed dynamics. A trapping region for $f$ is an open set $U$ with $f(U)\subset U$; the associated Conley attractor is $\bigcap_j f^j(U)$, and it is called strange when it is topologically transitive (some orbit is dense in it) and contains a non-periodic orbit with sensitive dependence. The rigorous landmark is Benedicks–Carleson (1991): for $b>0$ small enough, a positive measure set of $a$-values near $2$ gives a strange attractor on which a distinguished orbit satisfies the Collet–Eckmann condition, i.e. the norms of its forward derivatives grow exponentially. Mora–Viana (1993) extended this to Hénon-like families — $C^3$ perturbations of the form $(x,y)\mapsto(1-ax^2,0)$ of size $O(b^t)$ satisfying suitable determinant and derivative bounds. In parallel, Newhouse (1974) showed that generic surface diffeomorphisms can have infinitely many sinks (attracting periodic orbits); Robinson (1983) and later Gonchenko–Shilnikov–Turaev developed this via persistence of homoclinic tangencies, where a stable and an unstable manifold of a periodic point meet tangentially, using Baire-category and thickness arguments. Benedicks–Palmisano (2023) combined the constructive parameter-exclusion method of Benedicks–Carleson with Thunberg's unfolding of periodic attractors in the one-dimensional quadratic family and its Hénon analogue by Ures, to prove, without Baire category: (Theorem 1.3) for every Hénon-like family there is $b_0>0$ such that for each $0<b<b_0$ and each $k\ge 1$, a positive one-dimensional measure set of $a$-values carries at least $k$ sinks and one strange attractor; (Theorem 1.4) for each $0<b<b_0$ sufficiently small, all $a$ in some set $A_{\infty,b}$ carry infinitely many coexisting sinks — an assertion of membership in a parameter set, with no positive-measure claim, in contrast to Theorem 1.3; (Theorem 1.5) positive two-dimensional measure statements for the original Hénon family for the finite-$k$ coexistence, together with the existence (not positive measure) of Hénon parameters with infinitely many attractive periodic orbits; and (Theorem 1.6, confirming numerics of Curry, 1979) a positive two-dimensional measure set of parameters whose Hénon maps have two coexisting strange attractors. The mechanism fundamentally uses strong dissipation: for $b$ small, $f_{a,b}$ is a small singular perturbation of the one-dimensional quadratic map $x\mapsto 1-ax^2$, and the inductive distortion control of the parameter-exclusion algorithm is only proved under $b<b_0$. For the two-strange-attractor theorem (Theorem 1.6) the authors flag the need for distortion control of the $b$-derivatives — uniformity of the parameter-exclusion construction in the dissipation parameter $b$ — in the preprint (arXiv:1811.00517 v3), where the cross-reference to that estimate is unresolved; the sentence was removed in the published version and no such estimate was added, so the Fubini step that concludes the positive two-dimensional measure claim of Theorem 1.6 relies on a uniformity in $b$ that is not established in either version. Already Hénon's original parameter values $(a,b)=(1.4,\,0.3)$ lie well outside any regime where a positive-measure strange attractor (or sink coexistence) has ever been established, and near $b=1$ the map becomes almost area-preserving, where attracting orbits cannot exist at $b=1$ itself; how far the coexistence phenomena reach in $b$ is genuinely unknown. The results are framed as steps in the Palis conjecture program on the denseness of finitude of attractors for dissipative systems.

Problem Statement

The scientific objective is to determine the parameter extent and the quantitative size of positive-measure coexistence of attractors in the classical Hénon family $f_{a,b}(x,y)=(1-ax^2+y,\,bx)$, addressing the limitations left by Benedicks–Palmisano (2023) — the unspecified small-dissipation threshold, the absence of quantitative measure bounds, and the author-flagged missing distortion estimates — through three linked components, each of which belongs to the objective. (1) Parameter-range extension: the theorems establish, for some unspecified $b_0>0$, positive (one- and two-dimensional) Lebesgue measure parameter sets with (i) at least $k\ge 1$ attractive periodic orbits coexisting with one Benedicks–Carleson/Mora–Viana-type strange attractor and (iii) two coexisting strange attractors, but only for $0<b<b_0$; for (ii) infinitely many attractive periodic orbits the paper proves only the existence of parameters (Theorems 1.4 and 1.5(b)), with no positive-measure claim. Determine how far in $b$ these coexistence results extend — up to which explicit dissipation levels: resolving this component requires exhibiting an explicit dissipation threshold $b_*$, with all accompanying constants made explicit, such that the stated coexistence conclusions hold for every $0<b<b_*$, so that the extension is certified by a determinate range rather than by comparison with the source's unspecified $b_0$; the full classical domain $0<a<2$, $0<b<1$ is the aspirational endpoint rather than a prerequisite (noting that at $b=1$ the map is area-preserving and attracting orbits cannot exist, so the maximal attainable range is itself part of the question). (2) Quantitative size: prove effective lower bounds on the Lebesgue measures of the good parameter sets $A_{k,b}$ and $\mathcal{A}$ — that is, in the notation of Benedicks–Palmisano (2023), the set of $a$-values (for fixed $b$) for which $f_{a,b}$ has at least $k$ attractive periodic orbits and one strange attractor (Theorem 1.3), and the positive two-dimensional measure set of parameters $(a,b)$ for which $f_{a,b}$ has two coexisting strange attractors (Theorem 1.6, where it is denoted $A$) — as explicit functions of $b$ and the other constants when the set is one-dimensional (the fixed-$b$ sets $A_{k,b}$), and as explicit numerical constants independent of $b$ when the set is two-dimensional (the set $A$ in the $(a,b)$-plane); for the set $A_{\infty,b}$ of $a$-values for which $f_{a,b}$ has infinitely many coexisting attractive periodic orbits (Theorem 1.4), whose positive measure is not established in the paper, the corresponding task is to prove positive measure in the first place and only then to quantify it. (3) Completeness of the two-strange-attractor argument: determine whether the positive two-dimensional measure claim of Theorem 1.6 can be completely justified.

References

1. Michael Benedicks; Liviana Palmisano. Coexistence phenomena in the Hénon family. Bull. Braz. Math. Soc. (N.S.) 54(3), Paper No. 42 (2023). DOI: 10.1007/s00574-023-00345-9. https://doi.org/10.1007/s00574-023-00345-9 (preprint arXiv:1811.00517, https://arxiv.org/abs/1811.00517)
2. Michel Hénon. A two-dimensional mapping with a strange attractor. Comm. Math. Phys. 50, 69–77 (1976). DOI: 10.1007/BF01608556. https://doi.org/10.1007/BF01608556
3. Sheldon E. Newhouse. Diffeomorphisms with infinitely many sinks. Topology 13, 9–18 (1974). DOI: 10.1016/0040-9383(74)90034-2. https://doi.org/10.1016/0040-9383(74)90034-2
4. Michael Benedicks; Lennart Carleson. On iterations of 1-ax^2 on (-1,1). Ann. of Math. (2) 122, 1–25 (1985). DOI: 10.2307/1971367. https://doi.org/10.2307/1971367
5. Michael Benedicks; Lennart Carleson. The dynamics of the Hénon map. Ann. of Math. (2) 133, 73–169 (1991). DOI: 10.2307/2944326. https://doi.org/10.2307/2944326
6. Leonardo Mora; Marcelo Viana. Abundance of strange attractors. Acta Math. 171, 1–71 (1993). DOI: 10.1007/BF02392766. https://doi.org/10.1007/BF02392766
7. Hans Thunberg. Unfolding of chaotic unimodal maps and the parameter dependence of natural measures. Nonlinearity 14, 323–337 (2001). DOI: 10.1088/0951-7715/14/2/308. https://doi.org/10.1088/0951-7715/14/2/308
8. Raúl Ures. On the approximation of Hénon-like attractors by homoclinic tangencies. Ergodic Theory Dynam. Systems 15, 1223–1229 (1995). DOI: 10.1017/S0143385700009895. https://doi.org/10.1017/S0143385700009895
9. Clark Robinson. Bifurcation to infinitely many sinks. Comm. Math. Phys. 90, 433–459 (1983). DOI: 10.1007/BF01206892. https://doi.org/10.1007/BF01206892
10. James H. Curry. On the Hénon transformation. Comm. Math. Phys. 68, 129–140 (1979). DOI: 10.1007/BF01418124. https://doi.org/10.1007/BF01418124
11. Qiudong Wang; Lai-Sang Young. Toward a theory of rank one attractors. Ann. of Math. 167, 349–480 (2008). DOI: 10.4007/annals.2008.167.349. https://doi.org/10.4007/annals.2008.167.349
12. Jacob Palis, Jr. A global view of dynamics and a conjecture on the denseness of finitude of attractors. Astérisque 261, 335–347 (2000). DOI: 10.24033/ast.478. https://doi.org/10.24033/ast.478
13. Pierre Berger. Generic family with robustly infinitely many sinks. Invent. Math. 205, 121–172 (2016). DOI: 10.1007/s00222-015-0632-6. https://doi.org/10.1007/s00222-015-0632-6
14. Pierre Berger; Sylvain Crovisier; Enrique Pujals. Germ-typicality of the coexistence of infinitely many sinks. Adv. Math. 407, Article 108528 (2022). DOI: 10.1016/j.aim.2022.108528. https://doi.org/10.1016/j.aim.2022.108528
15. Shin Kiriki; Ming-Chia Li; Teruhiko Soma. Coexistence of invariant sets with and without SRB measures in Hénon family. Nonlinearity 23(9), 2253–2269 (2010). DOI: 10.1088/0951-7715/23/9/010. https://doi.org/10.1088/0951-7715/23/9/010
16. Michael Benedicks; Marco Martens; Liviana Palmisano. Newhouse Laminations. Preprint arXiv:1811.00617 (2018). arXiv: 1811.00617. https://arxiv.org/abs/1811.00617
17. Liviana Palmisano. Coexistence of non-periodic attractors. Preprint arXiv:1903.01446 (2019). arXiv: 1903.01446. https://arxiv.org/abs/1903.01446
18. Marco Martens; Liviana Palmisano. Wild Sets with Collet–Eckmann Points and Infinitely Many Sinks: Stability and Coexistence. Preprint arXiv:2511.12926 (2025). arXiv: 2511.12926. https://arxiv.org/abs/2511.12926
19. Yiheng Dong; Marco Martens; Liviana Palmisano. Introduction (chapter of One-Dimensional Dynamics, Lecture Notes of the Unione Matematica Italiana, 2025), 1–18. DOI: 10.1007/978-3-031-94882-4_1. https://doi.org/10.1007/978-3-031-94882-4_1
20. Alex Blumenthal; Yun Yang. Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps. Ann. Inst. H. Poincaré C Anal. Non Linéaire 39(2), 419–455 (2022). DOI: 10.4171/aihpc/11. https://doi.org/10.4171/aihpc/11