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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.

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.

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.

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.