# Is there a canonical notion of principal divisor on a discrete dynamical system?

I hope this question is well-posed.

Let (X, f) be a discrete dynamical system such that every x in X has finite period, i.e. there is some n such that f^n(x) = x. Let Div(X) be the free abelian group on the orbits of X. When X is a nonsingular algebraic curve over the algebraic closure of a finite field k and f is the Frobenius map, Div(X) is naturally isomorphic to the group of fractional ideals of k(X) (at least, I think; correct me if I'm wrong). There is a distinguished subgroup Prin(X) consisting of the preimage of the principal ideals, and Div(X)/Prin(X) is the divisor class group.

Is there a canonical definition of Prin(X) for general dynamical systems? If not, how much extra structure does X need to have for a construction like this to make sense and give some kind of useful information about X?

The case I'm interested in is that X is the set of aperiodic closed walks on a finite graph with a distinguished point and f moves the distinguished point.

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It seems like a problem for a general discrete dynamical system would be the identicalness of the points. In the case of an algebraic curve over the algebraic closure of a finite field, the points are endowed with extra structure from the notion of functions on the set, which is exactly where Prin(X) comes from. At least in your case, the discrete set still has a notion of 'closeness' that I guess might lend enable something like Prin(X), but I don't see any candidate. – Greg Muller Oct 21 '09 at 5:40