Problem: you can be the curl of a field without necessarily having your integral curves self intersect.
However, since the helmholtz decomposition (even in n > 3 dimensions) decomposes into a gradient of something and a vector field that’s divergenceless, and since divergenceless fields are volume preserving, if we assume that the possibilities must be in a finite volume, I think the Poincare recurrence theorem implies that for any set, almost every (as in, all but a volume 0 subset) point in the set returns to it. So, pretty sure you get that you get arbitrarily close to a cycle. Then if you have continuous preferences you can appeal to being money pumped? It’s close enough for me to think there’s something there, and that it’s useful in practice.
Problem: you can be the curl of a field without necessarily having your integral curves self intersect.
However, since the helmholtz decomposition (even in n > 3 dimensions) decomposes into a gradient of something and a vector field that’s divergenceless, and since divergenceless fields are volume preserving, if we assume that the possibilities must be in a finite volume, I think the Poincare recurrence theorem implies that for any set, almost every (as in, all but a volume 0 subset) point in the set returns to it. So, pretty sure you get that you get arbitrarily close to a cycle. Then if you have continuous preferences you can appeal to being money pumped? It’s close enough for me to think there’s something there, and that it’s useful in practice.