and k − − 1 is taken modulo N. In analogy to molecular motion, the system is time-reversible. Indeed, if balls would move clockwise (instead of counterclockwise) and marked points changed color upon entering them (instead of leaving), the motion would be equivalent, except going backward in time. Moreover, the evolution of η η k ( t ) is periodic, where the period is at most 2 N . (After N steps, each ball visits all M marked points and changes color by a factor ( − − 1 ) M .) Periodicity of the Kac ring is a manifestation of more general Poincaré recurrence.