Mathematical Sciences: Scales of Functions and Applications in Ergodic Theory

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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.