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.