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](http://www.springerlink.com/content/xr176777g1832x8v/). 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.



Is there a categorical treatment of dynamical systems?
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 $R[\mathcal S]$.


