  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.
