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Question: does this current definition of dynamical system generalise to dynamical systems over higher geometric structures such as smooth infinity-stacks, which are not sets in a cohesive type theory context?
Currently, the page (which is a little thin) states nothing but a smooth $\mathbb{R}^1$-group action. One would also want to refer to $\mathbb{Z}$-actions as dynamical systems, at times. Generally, one can of fix any group object $G$ encoding the desired type of (space-)time evolution and then regard $G$-action objects as dynamical systems in the given ambient category.
At this point this is really just “transformation group theory” with an interpretation of “dynamics” tacked onto it, dictated by the intended application.
Now, group actions, of course, generalize far and wide, in particular to infinity-actions of $\infty$-group stacks on $\infty$-stacks of any sort. Yes.
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