Saari's Conjecture is True for Generic Vector Fields

Saari's homographic conjecture for general masses in planar three-body problem under Newton potential and a strong force potential

Saari's homographic conjecture claims that, in the N-body problem under the homogeneous potential, $U=α^{-1}\sum m_i m_j/r_{ij}^α$ for $α\ne 0$, a motion having constant configurational measure $μ=I^{α/2}U$ is homographic, where $I$ represents the moment of inertia defined by $I=\sum m_i m_j r_{ij}^2/\sum m_k$, $m_i$ the mass, and $r_{ij}$ the distance between particles. We prove this conjecture for general masses $m_k>0$ in the planar three-body problem under Newton potential ($α=1$) and a strong force potential ($α=2$).

Super-hyperbolic orbits and non-collision singularities in a four-body problem

In this paper, we prove the existence of super-hyperbolic orbits in four-body problem, which solves a conjecture of Marchal-Saari. We also prove the existence of noncollision singularities in the same model, which solves a conjecture of Anosov. Moreover, the two type of solutions coexist for certain mass ratios. We also make a conjecture on the classification of final motions of N-body problem, in which the superhyperbolic type of orbits is one of the building blocks.

从拉格朗日三角到八边形编队：马歇尔猜想的证明

From the Lagrange Triangle to the Figure Eight Choreography: Proof of Marchal's Conjecture

本文证明了在等质量三体问题中，拉格朗日等边三角形解最对称的延续类包含摩尔发现的八边形编队，解决了马歇尔1999年提出的猜想。

The n-body problem in spaces of constant curvature

We generalize the Newtonian n-body problem to spaces of curvature k=constant, and study the motion in the 2-dimensional case. For k>0, the equations of motion encounter non-collision singularities, which occur when two bodies are antipodal. This phenomenon leads, on one hand, to hybrid solution singularities for as few as 3 bodies, whose corresponding orbits end up in a collision-antipodal configuration in finite time; on the other hand, it produces non-singularity collisions, characterized by finite velocities and forces at the collision instant. We also point out the existence of several cla…

Bowen's entropy-conjugacy conjecture is true up to finite index

For a topological dynamical system consisting of a continuous map f, and a (not necessarily compact) subset Z of X, Bowen (1973) defined a dimension-like version of entropy, h_X(f,Z). In the same work, he introduced a notion of entropy-conjugacy for pairs of invertible compact systems: the systems (X,f) and (Y,g) are entropy-conjugate if there exist invariant Borel subsets X' of X and Y' of Y such that h_X(f,X\setminus X') < h_X(f,X), h_Y(g,Y \setminus Y') < h_Y(g,Y), and (X',f|_{X'}) is topologically conjugate to (Y',g|_{Y'}). Bowen conjectured that two mixing shifts of finite type are entrop…