Mathematical Sciences: Scales of Functions and Applications in Ergodic Theory

该项目的重点是数学研究的问题， 动力系统 特别是，将在台球方面开展工作 多边形边界的系统。 一个目标是确定 无论是对所有或几乎所有无理直角三角形， 一般对称轨道必须是周期性的。 相交的轨道 垂直的边界称为对称。 工作将 也可以在检查合理的周期方向， 有理三角形 众所周知，每一个理性周期 方向必须是从 三个角度。 据信，一个匡威的陈述也 虽然不是普遍的。 A的适当条件 匡威有待进一步研究。 其他工作将考虑弗斯滕伯格的信件， 递归，线性块增长的最小符号流， 最小化由替换引起的流量。

This project focuses mathematical research on problems of dynamical systems. In particular, work will be done on billiard systems with polygonal boundaries. One goal is to determine whether for all, or almost all, irrational right triangles, a generic symmetric orbit must be periodic. Orbits which intersect the boundary perpendicularly are called symmetric. Work will also be done in examining rational periodic directions in rational triangles. It is known that every rational periodic direction must be a multiple of a fixed value computed from the three angles. It is believed that a converse statement also holds, although not universally. The proper conditions for a converse are still to be worked out. Other work will consider Furstenberg's correspondence and recurrence, minimal symbolic flows with linear block growth and minimal flows arising from substitutions.