论文标题
呼吸圆台球中的混乱运动
Chaotic motion in the breathing circle billiard
论文作者
论文摘要
我们考虑了圆形台球内部具有定期移动边界的圆形粒子的自由运动，并假设粒子与边界的碰撞是弹性的，因此不会保留粒子的能量。众所周知，如果边界的运动足够规律，则由于存在不变曲线，能量将被界定。我们表明，在移动边界的规律性假设下，粒子的运动可能是混乱的。更确切地说，我们表明存在一类函数，描述了边界的运动的运动，而台球地图则接受了带正值熵的不变概率度量。证明依赖于基于奥布里·妈妈理论的变异技术。
We consider the free motion of a point particle inside a circular billiard with periodically moving boundary, with the assumption that the collisions of the particle with the boundary are elastic so that the energy of the particle is not preserved. It is known that if the motion of the boundary is regular enough then the energy is bounded due to the existence of invariant curves. We show that it is nevertheless possible that the motion of the particle is chaotic, also under regularity assumptions for the moving boundary. More precisely, we show that there exists a class of functions describing the motion of the boundary for which the billiard map admits invariant probability measures with positive metric entropy. The proof relies on variational techniques based on Aubry-Mather theory.
