In the study of combinatorial objects and actions, such as standard Young tableaux under promotion, a typical sequence is to first understand the order of the action, then prove the cyclic sieving phenomenon of Reiner-Stanton-White (if it holds) and find results about the homomesy phenomenon of Propp-Roby, in which the orbit-average of a statistic on an object equals the global average. This program has been very successful. One difficulty, though, is that most of the objects that behave nicely in these respects are, in some sense, planar. These properties will often still hold for slightly 3-dimensional objects in cases where they may be mapped bijectively to planar objects. But once such maps are no longer possible, predictable orders, cyclic sieving, and homomesy typically do not occur.

In this talk, we introduce a new concept of resonance on discrete dynamical systems. This concept formalizes the observation that, in various combinatorially-natural cyclic group actions, orbit cardinalities are all multiples of divisors of a fundamental frequency. Resonance is a first step in understanding dynamics on more complicated combinatorial objects, such as plane partitions, since it is an analogue of an action having a nice order. We present these ideas via concrete examples and theorems and also give several open problems. This talk is based on joint papers with Kevin Dilks, Oliver Pechenik, and Nathan Williams.