Emergent spacetime and the ergodic hierarchy

Various diagnostics of the emergence of an arrow of time in the bulk description of a holographic theory have been proposed, including the decay of some real time correlation functions and the appearance of type III${}_1$ von Neumann algebras carrying half-sided modular inclusions. This note puts forward a close parallel between these diagnostics and a quantum formulation of the ergodic hierarchy of dynamical systems. Theories with an emergent spacetime appear to sit near the top of this hierarchy.