A abstract measure-preserving system consists of a space with a discrete-time dynamics and a time-invariant probability distribution.
Such systems provide the broadest setting for asking: What kinds of long-run randomness can be generated by dynamics? This is the province of ergodic theory.
The most thorough kind of answer to this question is phrased in terms of maps between systems. If a measure-preserving system exhibits a certain behaviour, we would like to find an equivariant map to another one whose special structure `explains' where the behaviour comes from.
This talk will survey some history, some recent progress, and some future directions in this subject. It will focus on the entropy of a measure-preserving system, which quantifies the unpredictability of future observations of the system given a record of observations into the distant past. I will cover some old and some quite new theorems which turn possible behaviours of the entropy into maps that account for it. I will finish with a sketch of a more modern setting where the analogous theory is still in its infancy: the study of actions of higher-rank free groups. Such actions arise from models of randomness distributed across a large disordered network, rather than randomness which emerges linearly with an evolution in time.