The remaining parameter gap in the Devaney–Nitecki conjecture: transversal homoclinic points of the area-preserving Hénon map for $-1<a\le-1+10^{-7}$

The area- and orientation-preserving Hénon map — in the classical parametrization $H_a(x,y)=(a-x^2-y,\,x)$, affinely conjugate to Hénon's original $F_a(x,y)=(1-ax^2+y,\,-x)$ — is the paradigmatic two-dimensional conservative quadratic map. Devaney and Nitecki (1979) conjectured that for every $a>-1$ (the parameter range in which the map has a hyperbolic fixed point) the stable and unstable manifolds of that fixed point intersect transversally, i.e. the map has a transversal homoclinic point and hence a horseshoe with chaotic symbolic dynamics. Successive rigorous results reduced the verified range from $a>5+2\sqrt{5}$ (Devaney–Nitecki), through $a\ge-0.3916$ (Fontich) and $a\ge-0.866360$ (Kirchgräber–Stoffer, computer-assisted), to the computer-assisted result of Antognini and Stoffer (2009), which is claimed to cover all $a>-1+10^{-7}$. This leaves open, conditional on that claimed coverage, the sliver $(-1,-1+10^{-7}]$ adjacent to the parabolic degeneration at $a=-1$, where the separatrix splitting becomes exponentially small and the validated-numerics constants degenerate. The problem is to prove, or disprove, transversal intersection of the invariant manifolds on this remaining interval, thereby completing the full conjecture provided the coverage for all $a>-1+10^{-7}$ reported by Antognini and Stoffer (2009) is correct; the peer-reviewed computer-assisted certified range remains $a\ge-0.866360$ (Kirchgräber–Stoffer 2006), so the stated sliver lies inside the publicly uncertified gap in any case.

Background

The Hénon maps are the quadratic diffeomorphisms of the plane with constant Jacobian; after scaling they take the form $(x,y)\mapsto(a-x^2+by,\,x)$ with $|b|\le1$ (Hénon, 1969, introduced the area-preserving member as a model problem for conservative dynamics). The area- and orientation-preserving case $b=-1$ is the map $H_a(x,y)=(a-x^2-y,\,x)$ studied here; its Jacobian determinant is identically $1$. (Hénon's original parametrization $F_a(x,y)=(1-ax^2+y,\,-x)$ is affinely conjugate to $H_a$ for every $a\ne0$, with the same parameter value.) The fixed points of $H_a$ are the points $(x,x)$ with $x^2+2x-a=0$, so two fixed points exist exactly for $a>-1$, and they merge at $a=-1$ into a single parabolic fixed point $(-1,-1)$ whose Jacobian has a double eigenvalue $1$. For every $a>-1$ the fixed point $z^{(1)}=(-1-\sqrt{1+a},\,-1-\sqrt{1+a})$ is hyperbolic (Jacobian eigenvalues $\lambda$ and $1/\lambda$ with $0<\lambda<1$), and it possesses one-dimensional stable and unstable manifolds (invariant curves consisting of points whose forward, respectively backward, iterates converge to $z^{(1)}$). A transversal homoclinic point is a point lying on both manifolds at which they cross at a nonzero angle; by the Smale–Birkhoff homoclinic theorem such a point forces a horseshoe, shift dynamics on infinitely many symbols, and positive topological entropy — the strongest standard certificate of deterministic chaos. Devaney and Nitecki (1979) constructed such dynamics for $a>(5+2\sqrt{5})(1+|b|)^2/4$ and conjectured that in the conservative case $b=-1$ a transversal homoclinic point exists for all $a>-1$. The conjecture is difficult near $a=-1$ for a structural reason: parameterizing the map by the stable eigenvalue $p\in(0,1)$ of $z^{(1)}$ (with $p\to1$ as $a\to-1$), the map is, for $p$ close to $1$, a discretization with step $h\propto 1-p\to0$ of an integrable flow with a separatrix; the angle at which the discrete stable and unstable separatrices split is then beyond all orders in $h$ (exponentially small), a phenomenon analyzed asymptotically by Tovbis–Tsuchiya–Jaffé (1998) and, for the parabolic map at $a=-1$ itself, by Gelfreich–Sauzin (2001) via Borel summation (a resummation technique that assigns values to divergent asymptotic series). Rigorous progress has come from two sides: analytic methods — Fontich (1990), using graph transforms (an iterative construction of invariant manifolds as limits of graphs), proved transversal intersection for $a\ge-0.3916$, and Brown (1995), via curvature estimates, for $a>0$ — and validated numerics, i.e. computations in interval arithmetic, where every quantity is enclosed in a machine-representable interval with outward (directed) rounding so that results are rigorously guaranteed: Kirchgräber–Stoffer (2006) introduced an a posteriori shadowing criterion (a technique that turns an approximate orbit into a nearby true orbit with rigorous error bounds) that converts a computed finite-support pseudo-orbit of the associated bi-infinite sequence equation into a certified hyperbolic homoclinic orbit, proving the result by hand for $a\ge17/64$ and, with interval arithmetic, for $a\ge-0.866360$. Antognini and Stoffer (2009) reported that an elaborated computer-assisted version of this pipeline (polynomial pseudo-orbit expansion in the parameter, interval-arithmetic bounds on the residual $r$ and on the inverse $K^{-1}$ of the linearized operator, checked against the a posteriori inequality $8\|K^{-1}r\|\,\|K^{-1}\|<1$ on each subinterval) establishes transversal intersection for all $a>-1+10^{-7}$, leaving only the interval adjacent to $-1$ unproved.

Problem Statement

For the area- and orientation-preserving Hénon map $H_a:\mathbb{R}^2\to\mathbb{R}^2$, $H_a(x,y)=(a-x^2-y,\,x)$ — Hénon's original parametrization $F_a(x,y)=(1-ax^2+y,\,-x)$ is affinely conjugate to $H_a$ for every $a\ne0$, with the same parameter value — let $z^{(1)}=(-1-\sqrt{1+a},\,-1-\sqrt{1+a})$ be the fixed point that is hyperbolic for every $a>-1$, whose stable eigenvalue $p\in(0,1)$ tends to $1$ as $a\to-1^+$. Prove, or disprove, that for every parameter value in the remaining interval

$-1<a\le-1+10^{-7}$

the stable and unstable manifolds of $z^{(1)}$ intersect transversally — that is, $z^{(1)}$ has a transversal homoclinic point. Equivalently, in the normal form of Kirchgräber–Stoffer (coordinates centered at $z^{(1)}$ and rescaled), the bi-infinite sequence equation $G(x)=0$ with $(G(x))_n=p\,x_{n+1}-(1+p^2)\,x_n+x_n^2+p\,x_{n-1}$ has a hyperbolic solution homoclinic to the origin, i.e. $x_n\to0$ as $n\to\pm\infty$. Combined with the previously established results — and provided the computer-assisted coverage for all $a>-1+10^{-7}$ reported by Antognini and Stoffer (2009) is correct — this would complete the Devaney–Nitecki conjecture for the entire half-line $a>-1$; since the peer-reviewed certified range remains $a\ge-0.866360$ (Kirchgräber–Stoffer 2006), the stated sliver lies inside the publicly uncertified gap whether or not that reported coverage holds. A disproof consists of a rigorous demonstration that for some $a^*$ in the interval the two invariant manifolds fail to intersect transversally; a proof must cover every parameter in the interval, not merely a subset.

References

1. Devaney, R.; Nitecki, Z. Shift automorphisms in the Hénon mapping. Communications in Mathematical Physics 67 (1979) 137–146. DOI: 10.1007/BF01221362. https://doi.org/10.1007/BF01221362
2. Hénon, M. Numerical study of quadratic area-preserving mappings. Quarterly of Applied Mathematics 27 (1969) 291–312. DOI: 10.1090/qam/253513. https://doi.org/10.1090/qam/253513
3. Fontich, E. Transversal homoclinic points of a class of conservative diffeomorphisms. Journal of Differential Equations 87 (1990) 1–27. DOI: 10.1016/0022-0396(90)90012-E. https://doi.org/10.1016/0022-0396(90)90012-E
4. Brown, Ray. Horseshoes in the measure-preserving Hénon map. Ergodic Theory and Dynamical Systems 15 (1995) 1045–1059. DOI: 10.1017/S0143385700009780. https://doi.org/10.1017/S0143385700009780
5. Stoffer, Daniel; Palmer, Kenneth J. Rigorous verification of chaotic behaviour of maps using validated shadowing. Nonlinearity 12 (1999) 1683–1698. DOI: 10.1088/0951-7715/12/6/316. https://doi.org/10.1088/0951-7715/12/6/316
6. Tovbis, Alexander; Tsuchiya, Masa; Jaffé, Charles. Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Hénon map as an example. Chaos 8 (1998) 665–681. DOI: 10.1063/1.166349. https://doi.org/10.1063/1.166349
7. Gelfreich, Vassili; Sauzin, David. Borel summation and splitting of separatrices for the Hénon map. Annales de l'Institut Fourier 51 (2001) 513–567. DOI: 10.5802/aif.1831. https://doi.org/10.5802/aif.1831
8. Kirchgräber, Urs; Stoffer, Daniel. Transversal homoclinic points of the Hénon map. Annali di Matematica Pura ed Applicata 185 (2006) S187–S204. DOI: 10.1007/s10231-004-0142-4. https://doi.org/10.1007/s10231-004-0142-4
9. Yamaguchi, Y.; Tanikawa, K. Order of Appearance of Homoclinic Points for the Henon Map. Progress of Theoretical Physics 116 (2006) 1029–1049. DOI: 10.1143/ptp.116.1029. https://doi.org/10.1143/ptp.116.1029
10. Antognini, Francesco; Stoffer, Daniel. On the Chaotic Behaviour of the Hénon Map. AIP Conference Proceedings 1168 (2009) 936–938. DOI: 10.1063/1.3241637. https://doi.org/10.1063/1.3241637 (source paper for the open question)
11. Gelfreich, Vassili; Naudot, Vincent. Width of the Homoclinic Zone in the Parameter Space for Quadratic Maps. Experimental Mathematics 18 (2009) 409–427. DOI: 10.1080/10586458.2009.10129058. https://doi.org/10.1080/10586458.2009.10129058 (preprint: arXiv:0805.0324, https://arxiv.org/abs/0805.0324)
12. de Oliveira, Vitor M.; Ciro, David; Caldas, Iberê L. Dynamical trapping in the area-preserving Hénon map. Preprint (2019). arXiv:1907.11616. https://arxiv.org/abs/1907.11616
13. Arai, Zin. On Hyperbolic Plateaus of the Hénon Map. Experimental Mathematics 16 (2007) 181–188. DOI: 10.1080/10586458.2007.10128992. https://doi.org/10.1080/10586458.2007.10128992
14. Arai, Zin; Chen, Yi-Chiuan. More on the concept of anti-integrability for Hénon maps. Journal of Mathematical Physics 66 (2025) 102706. DOI: 10.1063/5.0280115. https://doi.org/10.1063/5.0280115