## Is there a categorical treatment of dynamical systems?

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Let $X$ be a set and $(T,\cdot)$ an abelian group. Is there a category of $T$-dynamical systems on $X$ which yields useful information about $X$ and $T$?

More precisely, is there a category whose objects are dynamical systems, i.e., $\phi:X \times T \to X$ such that $\phi(x,t + s) = (\phi(\phi(x,t),s)$? And if so,

what are morphisms $\phi \to \phi'$ of $T$-dynamical systems on $X$?

I assume that in general this would involve imposing structure on $X$: for example, a topology so that one could consider homotopically perturbing $\phi$. Mostly, I am interested in asking when two such dynamical systems may be considered equivalent, and what it would take to have functors from $T$-dynamical systems on $X$ to $T'$-dynamical systems on $X'$, and to have natural transformations of those functors, etc.

I promise I've done (some) homework by looking at this. But note that this document only provides candidates for equivalent dynamical systems which presumably only accounts for isomorphisms in the desired category rather than all morphisms.

This may be too basic a question for the folks here; in this case I will delete it.

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The usual thing in dynamics if to fix a topological semigroup $T$ and looks at the category whose objects are actions of $T$ on a compact spaces and with maps the $T$-equivariant continuous maps. Isomorphisms in this category are commonly called conjugacies and epimorphisms are called factor maps. – Benjamin Steinberg Aug 3 at 3:37

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Perhaps you might change the question slightly. Is there a category of T-dynamical systems on X which gives useful information on the systems? Whether or not it is non-trivial as a category is not really that important. For some $(X,T)$ it might be trivial (in some sense)... and that might correspond to a useful class of dynamics. As a dynamical system is a group action, there is always a category of T-sets, and you can restrict to the full subcategory of those whose underlying set is X. – Tim Porter Aug 3 at 5:05
Dear Tim, I've made the edits. Thank you for your suggestion. Benjamin, I was aware of factor maps, but those are too specific since they require topological notions. In general, that framework wouldn't even accomodate shift maps on infinite strings in finitely many symbols since the space isn't compact, right? – Vidit Nanda Aug 3 at 17:04

## 3 Answers

For every category $\mathfrak C$ and every monoid $\mathcal S$, you can define the category of $\mathcal S$-flows on $\mathfrak C$ as follows:

-- Objects are pairs $(X, \alpha:\mathcal S\to \mathrm{End}(X))$, where $\alpha$ is a monoid homomorphism (that is, multiplication in $\mathcal S$ becomes composition of morphisms in $\mathfrak C$ and the neutral element goes to identity);

-- Morphisms in the category between two objects $(X,\alpha)$ and $(X',\alpha')$ are just morphisms $\phi:X\to X'$ that commute with the actions, that is $\alpha'(s)\phi=\phi\alpha(s)$ for all $s\in \mathcal S$.

Clearly an isomorphism in this category is just a morphism that is an isomorphism in $\mathfrak C$.

This is standard and probably does not answer completely your question as you probably want a weaker form of equivalence than really isomorphism in the category of $\mathcal S$-flows. Anyway I think this is a good starting point.

Just to understand what this category is, I can give you some examples. If you consider $\mathbb N$-flows on the category of modules $\mathrm {Mod}(R)$ over a ring $R$, you just obtain the category of modules over the polynomial ring $R[X]$. In fact, it is a classical way of looking at modules over $R[X]$ as $R$-modules with a distinguished $R$-linear endomorphism acting on them, that is, discrete-time dynamical systems.

If you take $\mathbb Z$-flows you obtain the ring of Laurent polynomials $R[X^{\pm 1}]$. This is easily generalized to $\mathbb N^k$ and $\mathbb Z^k$, giving rise to polynomials in $k$ commuting variables. This point of view is generally adopted by K. Schmidt in his book "Dynamical Systems of Algebraic Origin". In fact, the general approach there is to study dynamical systems of the form $(G,\phi_1,\dots,\phi_k)$, where $G$ is a compact abelian topological group and $\phi_1,\dots,\phi_k$ are commuting topological automorphisms of $G$. This is the category of $\mathbb Z^k$-flows on the category of compact abelian groups. Via Pontryagin duality, this category can be seen to be dual to the category of $\mathbb Z^k$-flows on discrete Abelian groups, that, by what we said above, is exactly $\mathrm{Mod}(\mathbb Z[X_1^{\pm 1},\dots,X_k^{\pm 1}])$.

Generalizing more, you can easily prove that $\mathcal S$-flows on $\mathrm {Mod}(R)$ are the category of modules over the monoid ring $R[\mathcal S]$.

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An action of a group $T$ on a set $X$ defines the action groupoid $T\times X \rightrightarrows X$ (If $T$ is a semigroup then $T\times X \rightrightarrows X$ is a category). Thinking of dynamical systems this way suggests that morphisms are functors between action groupoids. If there is a topology on $T$ and $X$ and the action is continuous, then the action groupoid is a topological groupoid. You may then take your morphisms to be continuous functors. However, this is not the best one can do. A better notion of morphism between topological groupoid is that of a bibundle also known as a ‘Hilsum–Skandalis map.’ If you go down this road you end up with bicategories since the composition of bibundles is associative only up to isomorphism (see this paper by Christopher Schommer-Pries for a nice discussion of the issues).

If your dynamical systems are vector fields on manifolds then the morphisms are smooth maps intertwining the vector fields. That is, if $X$ is a vector field on $M$, $Y$ a vector field on $N$ then a morphism from $(M,X)$ to $(N, Y)$ is a smooth map $f:M\to N$ with $Y \circ f = Df \circ X$.

There is also a large body of literature on the category of labelled transition systems and on the category of Petrie nets...

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Note a map from a terminal object is a fixed point and a monomorphism is an invariant subsystem.  In the case where the category in question is manifolds and vector fields, a map from a circle with the constant vector field is a periodic orbit.   – Eugene Lerman Aug 4 at 0:27

See "J. de Vries, Topological transformation groups 1, A cat.approach" and similar works about topological group actions from a categorical perspective.

(http://oai.cwi.nl/oai/asset/12605/12605A.pdf or search for "Topological transformation groups" (any author) in http://repository.cwi.nl/).

THese is a account of it in the last article in Springer Lecture notes in MAth n. 540 (catagorical topology).

ABout homotopical question of Topological transformation groups see the Tammo Tom Dieck book (http://www.amazon.com/Transformation-Groups-Gruyter-Studies-Mathematics/dp/3110097451)

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