标题： 符号动力学与心形台球的周期轨道 显示英文标题

标题： Symbolic Dynamics and Periodic Orbits for the Cardioid Billiard

摘要： 通过引入二进制符号动力学研究了强混沌心形散射系统的周期轨道。 相应的分区被映射到一个拓扑上良序的符号平面。 在符号平面上，修剪前沿由进入或穿过尖角的轨道获得。 我们证明所有周期轨道对应于拉格朗日的极大值，并给出了长度码长为15以内的完整列表。 对称性约化是在符号序列的层次上完成的，周期轨道使用对称线进行分类。 我们证明存在无限多个在长度上积累的周期轨道族，并且所有其他几何短周期轨道族最终都会被修剪。 所有这些轨道都与起始和终止于尖角的有限轨道相关。 我们得到了柯尔莫哥洛夫-西奈熵的一个解析估计，并发现与数值计算值以及通过对周期轨道取平均得到的值有很好的一致性。 此外还研究了周期轨道的统计性质。 显示英文摘要

摘要： The periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics. The corresponding partition is mapped to a topological well-ordered symbol plane. In the symbol plane the pruning front is obtained from orbits running either into or through the cusp. We show that all periodic orbits correspond to maxima of the Lagrangian and give a complete list up to code length 15. The symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines. We show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned. All these orbits are related to finite orbits starting and ending in the cusp. We obtain an analytical estimate of the Kolmogorov-Sinai entropy and find good agreement with the numerically calculated value and the one obtained by averaging periodic orbits. Furthermore the statistical properties of periodic orbits are investigated.