The First Return Time Properties of an Irrational Rotation

If an ergodic system has positive entropy, then the ShannonMcMillan-Breiman theorem provides a relationship between the entropy and the size of an atom of the iterated partition. The system also has OrnsteinWeiss’ first return time property, which offers a method of computing the entropy via an orbit. We consider irrational rotations which are the simplest model of zero entropy. We prove that almost every irrational rotation has the analogous properties if properly normalized. However there are some irrational rotations that exhibit different behavior.