Suitable categorical framework for ergodicity

Construct a suitable category for generalized dynamical systems without uniqueness of trajectories that permits formulation of ergodicity through the categorical definition of ergodicity in the Markov category.

Background

The paper notes a possible connection between its measure-theoretic notion of ergodicity and the categorical definition of ergodicity considered in the cited work on Markov categories. Although ergodicity can be formulated in a Markov category, applying that framework to generalized dynamical systems in the present setting requires a suitable category that captures their non-unique trajectories and associated invariant measures.

References

Another important question is the relation to the categorical definition of ergodicity considered in . One can formulate the notion of ergodicity for systems in the Markov category, and therefore we can expect to obtain a version of ergodicity by applying this definition. However, to make this possible in the current setting, we need to formulate a suitable category, which is an interesting question in itself.

— Ergodicity of dynamical systems without uniqueness of orbits  (2609.11087 - Suda, 10 Sep 2026) in Section 5, Concluding Remarks