# Localizations of categories of dynamical systems

## 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.

1. My natural numbers include 0 ↩︎