Area-Preserving Surface Diffeomorphisms

We prove some generic properties for $C^r$, $r=1, 2, ..., \infty$, area-preserving diffeomorphism on compact surfaces. The main result is that the union of the stable (or unstable) manifolds of hyperbolic periodic points are dense in the surface. This extends the result of Franks and Le Calvez \cite{FL03} on $S^2$ to general surfaces. The proof uses the theory of prime ends and Lefschetz fixed point theorem

Convex central configurations for the n-body problem

AbstractWe give a simple proof of a classical result of MacMillan and Bartky (Trans. Amer. Math. Soc. 34 (1932) 838) which states that, for any four positive masses and any assigned order, there is a convex planar central configuration. Moreover, we show that the central configurations we find correspond to local minima of the potential function with fixed moment of inertia. This allows us to show that there are at least six local minimum central configurations for the planar four-body problem. We also show that for any assigned order of five masses, there is at least one convex spatial central configuration of local minimum type. Our method also applies to some other cases

Melnikov method and transversal homoclinic points in the restricted three-body problem

AbstractIn this paper we show, by Melnikov method, the existence of the transversal homoclinic orbits in the circular restricted three-body problem for all but some finite number of values of the mass ratio of the two primaries. This implies the existence of a family of oscillatory and capture motion. This also shows the non-existence of any real analytic integral in the circular restricted three-body problem besides the well-known Jacobi integral for all but possibly finite number of values of the mass ratio of the two primaries. This extends a classical theorem of Poincaré [10]. Because the resulting singularities in our equation are degenerate, a stable manifold theorem of McGehee [7] is used