The marked length spectrum of a closed, negatively curved Riemannian manifold records the lengths of closed geodesics. We can compare two Riemannian metrics using their marked length spectra, which can be done dynamically via the geodesic flow. This perspective has been extended to other geometric contexts, such as pairs of Anosov representations and actions on metric trees. In this talk, I will discuss a joint work with Stephen Cantrell in which we compare length functions of actions on CAT(0) cube complexes. These are non-positively curved spaces of combinatorial nature that generalize simplicial trees. The role of the geodesic flow is now played by a finite-state automaton, inspired by Calegari-Fujiwara’s work about word metrics on hyperbolic groups.