Partially hyperbolic dynamics in the 3-body problem

TL;DR

The paper develops a framework for symplectic blenders and demonstrates their robust topological instability in celestial-mechanics models. It introduces two abstract results—an explicit condition for local transitivity of a pair of twist maps on the annulus (and a cylinder skew-product extension)—and then applies them to construct symbolic and symplectic blenders in the 3-body problem and its restricted version. By embedding a partially hyperbolic setting via McGehee-type coordinates and analyzing scattering/return maps, it proves the existence of orbits with center-dynamics that are locally dense within a normally parabolic lamination, achieving a strong form of Arnold diffusion-like behavior. The results provide a systematic route to identify blenders in parametric Hamiltonian families and illustrate their global impact on dynamics, including robust transitivity and almost-dense center behavior in physically relevant models.

Abstract

We construct symplectic blenders for two classical Hamiltonian systems: the 3-body problem and its restricted version. We use these objects to show that both models exhibit a robust, strong form of topological instability. We do not assume any smallness conditions on the masses but require only that at least two of them are distinct. Our construction is based on two abstract results which might be of independent interest. The first one gives an explicit condition under which a given pair of twist maps of the cylinder generates a locally transitive iterated function system. The second one extends this result to certain cylinder skew-products.

Figures (8)

• Figure 1.1: Let $F_n=T_0^n\circ T_1$. In the left we show the image of a small rectangle $D$ under the map $F_n$ for $n\gg 1/\varepsilon$. In the right we show the image of $D$ under $F_{n_i}$, $i=1,2,3$ for $n_i\sim 1/\varepsilon$. In this regime the expansion/contraction is arbitrarily close to one. Moreover, if $\beta$ is sufficiently irrational it is possible to chose $n_i$ such that the union $\bigcup_{i}F_{n_i}(D)$ contains $D$.
• Figure 1.2: An ellipse with unit semimajor axis, eccentricity $\epsilon\in[0,1)$ and argument of the pericenter $g\in\mathbb T$. The position of the red point inside the ellipse is measured by the angle $\ell\in\mathbb T$.
• Figure 1.3: The primaries (i.e. the massive bodies) $q_0,q_1$ orbit around themselves, each of them describing an ellipse around the center of mass. The massless body $q$ moves influenced by their gravitational field.
• Figure 4.1: The strip $\Delta$ intersects only the vertical rectangles $V_1,V_2$. It intersects $V_2$ cleanly, i.e. $\hat{\Delta}_2\subset V_2$ but the intersection with $V_1$ is not clean, i.e. $\hat{\Delta}_1\nsubset V_1$. However, the smaller subset $\tilde{\Delta}_1$ is contained in $V_1$.
• Figure 5.1: The expanding (red) and contracting (blue) directions associated to the hyperbolic matrix $A_N$. Theses directions are approximately symmetric with respect to the $x$-axis and the angle between them is of order $\angle \sim \varepsilon/\chi^2$. The gray rectangle corresponds to the image of $D$ under $\phi_\chi$.

Theorems & Definitions (110)

• Theorem A
• Remark 1
• Definition 1.1
• Theorem B
• Theorem C
• Theorem D: Informal version
• Theorem D: Geometric version
• Theorem E
• Definition 3.1: Symbolic $cs$-blender
• Definition 3.2: Symbolic double blender