Joinings in Markov categories
Abstract
Dynamical systems theory primarily concerns the study of different forms of invariance, such as invariant sets, densities, measures, and observables. Ergodic systems are a special class of dynamical systems which are measure theoretically irreducible. In spite of this specialized property they dominate most discussions on ergodic theory because of the ergodic decomposition theorem. The viewpoint that any dynamical system is a composite of multiple ergodic components enables us to partition the face space into the basins of the different measures. The coexistence and mutual connections between these various coexisting subsystems are illuminated very effectively using the language of Category theory (CT). Some recent advancements have shown how the essence of measure theoretic dynamical systems can be captured through the formalism of Markov categories. This formalism captures the essential features of invariance and ergodicity through the language of limits and colimits. This article continues that formalism by studying the concept of joins using the language of spans and push-outs. A classical result is re-proven which establishes the connection between joins, ergodicity and mixing. In the process, the categorical language is refined to capture the notion of sub sigma-algebras and their invariance.