Abstract:
In this talk I will introduce the spatial isosceles three-body problem from the perspective of Symplectic Dynamics. For certain choices of mass ratio, angular momentum, and energy, the dynamics on the energy surface is equivalent to a Reeb flow on the tight three-sphere. We find a Hopf link formed by brake orbits, which spans an open book decomposition whose pages are annulus-like global surfaces of section. For large mass ratios, we show that the Hopf link is non-resonant, i.e., the rotation numbers of its components satisfy a non-resonance condition. In particular, the first return map to a page of the open book satisfies a twist condition, implying the existence of infinitely many periodic orbits. These orbits are distinguished by their projections to the Hill's region. We also find a positively measured subset of parameters, so that the first return map satisfies a rational condition that also implies infinitely many periodic orbits. Finally, we find the complete range of parameters for which the energy surface is strictly convex.
Biography:
Lei Liu obtained his Bachelor degree at Harbin Engineering University, school of science, in June 2015. Then he obtained his Ph.D. degree from the dynamical system group at Shandong University, school of mathematics, in June 2021. During these years, he was advised by Professor Xijun Hu in Shandong University. Now he is holding a postdoctoral position in Peking University, BICMR and his co-advisor is Professor Gang Tian in Peking University. His research is mainly about the Hamiltonian system, Maslov index theory and symplectic dynamics.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai