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)[^fn: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.

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.

[^fn:1]: My natural numbers include 0