Time-periodic behaviour in one- and two-dimensional interacting particle systems

We provide a class of examples of interacting particle systems on Zᵈ, for d∈\1,2\, that admit a unique translation-invariant stationary measure, which is not the long-time limit of all translation-invariant starting measures, due to the existence of time-periodic orbits in the associated measure-valued dynamics. This is the first such example and shows that even in low dimensions, not every limit point of the measure-valued dynamics needs to be a time-stationary measure.