Localizations of categories of dynamical systems

See also: Jade Master: Dynamical Systems With Category Theory? Yes!, This tweet by me.

Discrete dynamical systems

A discrete dynamical system \((S,T)\) consists of a set \(S\) and a time-step function \(T: S \to S\). It’s clear that this is exactly the same thing as a \(\mathbb{N}\)-set, i.e a set with an action of the monoid \((\mathbb{N},+,0)\)¹. A “morphism of discrete dynamical systems” is just the obvious thing, namely a map which preserves the action - an equivariant map.

For now, we will restrict ourselves to those dynamical systems with “time-reversible dynamics” - that is, those where the action \(T: S \to S\) is a bijection. The subcategory of such things inside \(\mathbb{N}-\mathsf{Set}\) is equivalent to \(\mathbb{Z}-\mathsf{Set}\) - a dynamical system is time-reversible if and only if the action of \(\mathbb{N}\) extends to an action of \(\mathbb{Z}\).

Now, we will be interested in localizations of the category \(\mathbb{Z}-\mathsf{Set}\). One important family of such localizations comes from the group homomorphisms \(\mathbb{Z} \to \mathbb{Z}/n\).

This homomorphism gives a functor \(\mathbb{Z}/n-\mathsf{Set} \hookrightarrow \mathbb{Z}-\mathsf{Set}\). In fact, this functor is fully faithful, and its image is exactly the subcategory of $n$-periodic dynamical systems - i.e those for which \(T^n(x) = x\) for all \(x \in S\). Moreover, this functor admits a left adjoint \(L_n\). It takes a dynamical system \((S,T)\) to \((S/\sim, \bar{T})\), where \(\sim\) is the equivalence relation generated by \(x \sim T^nx\) and \(\bar{T}\) is the induced map.

The order structure of this family of localizations is exactly the division order. By which I simply mean, if \(n | m\), then an $n$-periodic dynamical system is also $m$-periodic, and if every $n$-periodic dynamical system is $m$-periodic, then \(n|m\) (proof of the second implication: consider the dynamical systen \((\mathbb{Z}/m, +1)\)).

Using these, we can form a sort of “\(p\)-adic completion” of any dynamical system, as the limit \(\lim_n L_{p^n} S =: S^{\wedge}_p\). The functor \((-)^\wedge_p\) is a localization. The \(p\)-adic completion of the integers with translation action is exactly the $p$-adic integers (with translation action). Since each system \(S/p^n\) acquires a canonical action of \(\mathbb{Z}/p\), it would seem that probably \(S^\wedge_p\) acquires an action of the \(p\)-adics \(\mathbb{Z}^\wedge_p\).

However, to make sense of this, it’s probably best to work in a category of topological spaces with continuous group actions. I would guess that the \(\mathbb{Z}/p\) actions assemble to a unique continuous \(\mathbb{Z}_p^\wedge\) action, although I have not checked it.

The family of localizations \((-)^\wedge_p\) is jointly conservative on finite dynamical systems (since each orbit is an $n$-period for some \(n\)). However this fails in a predictable way in the infinite case, where we can’t distinguish between \((\mathbb{Z},+)\) and \((\hat{\mathbb{Z}},+)\). To remedy this, one would need some sort of “rationalization” of dynamical systems. However, there we run into the issue that the functor \(\mathbb{Q}-\mathsf{Set} \to \mathbb{Z}\mathsf{Set}\) is not fully faithful - a \(\mathbb{Q}\) set has a chosen “half action”, the action of \(1/2\) on \(S\), but this action is not necessarily uniquely determined by the action of \(1\).

Smooth dynamical systems

We take a somewhat unorthodox approach and let a smooth dynamical system be a smooth manifold \(M\) with an action of the Lie group \(\mathbb{R}\). (Again, we are looking at time-reversible systems). Given a discrete subgroup of \(\mathbb{R}\), the quotient \(\mathbb{R}/H\) is again a Lie group, and we obtain a fully faithful inclusion \(\mathbb{R}/H-\mathsf{Mdf} \hookrightarrow \mathbb{R}-\mathsf{Mdf}\).

The discrete subgroups of \(\mathbb{R}\) all have the form \(\lambda\mathbb{Z}\) for some \(\lambda\), and the quotient is always diffeomorphic to \(S^1\). As above, passing to this quotient corresponds to considering $λ$-periodic systems.

In this situation, there no longer exists a left adjoint, for annoying reasons. The universal property of the left adjoint, if written out, tells us it should take a manifold \(M\) to the quotient \(M/(\lambda\mathbb{Z})\) of the action by \(\lambda\mathbb{Z}\). Consider the normal additive action of \(\mathbb{R}\) on \(S^1 = \mathbb{R}/\mathbb{Z}\). If we take \(\lambda\) to be an irrational number, the orbits of the action are dense in \(S^1\), and the quotient is not even a topological manifold.

One would hope this problem can be solved by something like “derived manifolds”, but I haven’t looked into that yet.