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.