Abstract
In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way.
Highlights
The theory of autonomous dynamical systems has developed in different flavours according to which structure is assumed on the state space—with topological dynamics and ergodic theory as two prominent examples
These theories have followed parallel tracks and built multiple bridges. Both have grown successful tools from the interaction with information theory soon after its emergence, around the key invariants, namely the topological entropy and metric entropy, which are themselves linked by a variational theorem
This is a far-reaching programme, we show how category theory can help re-derive some non-trivial facts of dynamical systems
Summary
The theory of autonomous dynamical systems has developed in different flavours according to which structure is assumed on the state space—with topological dynamics (studying continuous maps on topological spaces) and ergodic theory (studying probability measures invariant under the map) as two prominent examples. Ornstein’s isomorphism theorem states that the monoid corresponding to the subcategory of Bernoulli shifts endowed with the tensor product is the nonnegative real numbers with the addition. It offers an alternative to the derivation of Shannon entropy as the unique quantity (up to a multiplicative constant) that satisfy a list of natural axioms [5,6] It complements the categorical interpretation of entropy by Baez et al [7], which characterizes the loss of Shannon entropy along an arrow in Prob as a functor to the category of one object with all reals as arrows. A natural follow-up would be to look for a categorical proof of those results This is a far-reaching programme, we show how category theory can help re-derive some non-trivial facts of dynamical systems
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